We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the 17th and 18th century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a "primary point of reference for understanding the eighteenth-century theories". Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own.Euler's use of infinite integers and the associated infinite products is analyzed in the context of his infinite product decomposition for the sine function. Euler's principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler.We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

The goal of this study was to investigate the molecular interaction between superantigens and the T-cell receptor (TCR). Using a quantitative polymerase chain reaction (PCR) to assess T-cell proliferation profiles, we found that SEB, SEC1, SEC2 and SEC3 expanded human T cells bearing V beta 3, V beta 12, V beta 13.2, V beta 14, V beta 15, V beta 17 and V beta 20. SEC2 and SEC3 have the additional ability to expand T cells bearing V beta 13.1, and their expansion of V beta 3 was markedly reduced compared to SEB and SEC1. Based on the activity of SEC1 mutants containing single amino acid substitutions, we concluded that the differential abilities of these native toxins to stimulate V beta 3 and V beta 13.1 was determined by the residue in position 26, located in the base of the SEC alpha 3 cavity. The SEC1 mutant, in which Val in position 26 was substituted with the analogous SEC2/SEC3 residue (Tyr), generated a V beta expansion profile that was indistinguishable from those generated by SEC2 and SEC3. Using these findings, the co-ordinates of a recently reported murine TCR beta-chain crystal structure, and other documented information, we propose a compatible molecular model for the interaction of SEC3 with the T-cell receptor. In this model complex, the complementarity-determining regions (CDRs) 1 and 2 and the hypervariable loop 4 of the V beta element contact SEC3 predominantly through residues in the alpha 3 cavity of the toxin. CDR3 of the beta chain is not involved in any toxin contacts. The proposed model not only includes contacts identified in previous mutagenesis studies, but is also consistent with the ability of tyrosine and valine in position 26 to differentially affect the expansion of V beta s 3 and 13.1 by the SEC superantigens.

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

Staphylococcus aureus isolates from bovine and ovine species produce unique molecular variants of type C staphylococcal enterotoxin (SEC). The SEC animal variants have greater than 98% amino acid sequence identity with SEC1, a human-associated SEC. The two SEC animal variants have been designated SEC bovine and SEC ovine according to their corresponding host species. We showed previously that these toxins induce quantitatively different levels of T-cell stimulation in several animal species. The present study compared the abilities of these closely related host-specific SEC variants to stimulate V␤-bearing T cells from bovine and human donors. All three toxins expanded human T cells bearing T-cell receptor V␤ elements (huV␤) 3, 12, 13.2, 14, 15, 17, and 20. However, SEC1 resulted in greater expansion of hyV␤12 than either SEC bovine or SEC ovine. In addition, bovine T cells proliferate in a V␤-dependent manner in response to these superantigens (SAgs). All three toxins induced the proliferation of bovine T cells bearing the previously sequenced V␤ element (boV␤) from the bovine T-cell clone BTB13 (boV␤BTB13). SEC1 and SEC ovine also were able to induce proliferation of bovine T cells bearing boV␤BTB35, which SEC bovine failed to stimulate. The species-specific differences in T-cell proliferation exhibited by these closely related SEC variants may reflect the evolutionary adaptation of S. aureus, presumably to increase its host range by the manipulation of the immune system in a host-specific manner.

. Cauchy, infinitesimals and ghosts of departed quantifiers, Mat. Stud. 47 (2017), 115-144.Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence.As case studies, we analyze the approaches of Craig Fraser and Jesper Lützen to Cauchy's contributions to infinitesimal analysis, as well as Fraser's approach toward Leibniz's theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser's interpretive framework.