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Emanuele Bottazzi

Publications and source records attributed to Emanuele Bottazzi.

17 recordsLinked to original sources
arXiv2026

Walking on the DARKSIDE

Large Language Models (LLMs) do not natively track the path of exclusions that a coherent discourse demands. When an input rests on a fabricated authority, a misapplied mechanism, or a surreptitious analogy, an unsteered LLM tends to engage with it as if it were well-posed, and this affects its generation. POLANYI++, an LLM-steering method that uses heuristics, ontologies and problem-solving methods for tacit-knowledge extraction, produces an Extended Knowledge Graph (XKG) in OWL2, but when a sophisticated nonsensical input is reified into the graph alongside the legitimate triples, it gets hardly detectable by automated reasoners, since the XKG is generated jointly with the wrong assumptions. We introduce DARKSIDE, a coherence-auditing method on top of POLANYI++. DARKSIDE formalises an explicit data structure of accumulated exclusions over discourse time, complemented by a warrant axis that classifies each named referent as Warranted, Unattested, Misattributed or Fabricated. The method is anchored in nine theoretical fragments unified under a shared deep frame of path integrity. The resulting DARKPOLANYI is evaluated as a steering layer over Gemini 3 on BSBench, a 100-item adversarial corpus of sophisticated-sounding nonsense across multiple domains, with Claude Sonnet 4.6 as an independent judge. DARKPOLANYI scores 1.89/2 mean versus 0.95/2 for the unsteered Gemini 3 Pro baseline; on the 97 cases with valid judgments in both arms, paired McNemar gives a paired bootstrap mean-diff = +0.92 (95% CI [+0.75, +1.08], p = 0.0001). The evidence supports an architectural claim: when an LLM forward pass is wrapped in an ontology-mediated auditing, structurally inevitable hallucination can be partially recovered. The XKG functions as the missing memory that LLMs lack, and the warrant axis as an epistemic firewall.

cs.AI↗
arXiv2025

History of Archimedean and non-Archimedean approaches to uniform processes: Uniformity, symmetry, regularity

We apply Nancy Cartwright's distinction between theories and basic models to explore the history of rival approaches to modeling a notion of chance for an ideal uniform physical process known as a fair spinner. This process admits both Archimedean and non-Archimedean models. Advocates of Archimedean models maintain that the fair spinner should satisfy hypotheses such as invariance with respect to rotations by an arbitrary real angle, and assume that the optimal mathematical tool in this context is the Lebesgue measure. Others argue that invariance with respect to all real rotations does not constitute an essential feature of the underlying physical process, and could be relaxed in favor of regularity. We show that, working in ZFC, no subset of the commonly assumed hypotheses determines a unique model, suggesting that physically based intuitions alone are insufficient to pin down a unique mathematical model. We provide a rebuttal of recent criticisms of non-Archimedean models by Parker and Pruss.

math.HO↗
arXiv2023

Flexible involutive meadows

We investigate a notion of inverse for neutrices inspired by Van den Berg and Koudjeti's decomposition of a neutrix as the product of a real number and an idempotent neutrix. We end up with an algebraic structure that can be characterized axiomatically and generalizes involutive meadows. The latter are algebraic structures where the inverse for multiplication is a total operation. As it turns out, the structures satisfying the axioms of flexible involutive meadows are of interest beyond nonstandard analysis.

math.LO↗
arXiv2021

Integration with filters

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part. These numbers can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations valid for real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.

math.LO↗
arXiv2021

Infinitesimals via Cauchy sequences: Refining the classical equivalence

A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy's sentiment that a null sequence "becomes" an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor's belittling of infinitesimals.

math.LO↗
arXiv2021

Describing limits of integrable functions as grid functions of nonstandard analysis

In functional analysis, there are different notions of limit for a bounded sequence of $L^1$ functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $L^1$ functions can be described in terms of its weak-$\star$ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson's framework of analysis with infinitesimals, we show that for every bounded sequence $\{z_n\}_{n \in \mathbb{N}}$ of $L^1$ functions there exists a function of a hyperfinite domain (i.e.\ a grid function) that represents both the weak-$\star$ and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward-backward parabolic equation.

math.FA↗
arXiv2020

A real-valued measure on non-Archimedean field extensions of $\mathbb{R}$

We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields $(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure ${m_L}$ turns out to be general enough to obtain a canonical measurable representative in $\mathbb{F}$ for every Lebesgue measurable subset of $\mathbb{R}$, moreover, the measure of the two sets is equal. In addition, $m_L$ it is more expressive than a class of non-Archimedean uniform measures. We focus on the properties of the real-valued measure in the case where $\mathbb{F}=\mathcal{R}$, the Levi-Civita field. In particular, we compare ${m_L}$ with the uniform non-Archimedean measure over $\mathcal{R}$ developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in $\mathcal{R}$. Recall that this result is false for the current non-Archimedean integration over $\mathcal{R}$. The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.

math.FA↗
arXiv2020

Infinite lotteries, spinners, and the applicability of hyperreals

We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei-Shelah model or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In the second part we will analyze two underdetermination theorems by Pruss and show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined.

math.HO↗
arXiv2020

Internality, transfer, and infinitesimal modeling of infinite processes

A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson's transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.

math.HO↗
arXiv2020

Spaces of measurable functions on the the Levi-Civita field

We introduce the $\mathcal{L}^p$ spaces of measurable functions whose $p$-th power is summable with respect to the uniform measure over the Levi-Civita field $\mathcal{R}$. These spaces are the counterparts of the real $L^p$ spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the $L^p$ spaces: for instance, the $\mathcal{L}^p$ spaces are not complete with respect to the $p$-norm. This motivates the study of the completions of the $\mathcal{L}^p$ spaces with respect to strong convergence, denoted by $\mathcal{L}_s^p$. It turns out that the $\mathcal{L}_s^p$ spaces are Banach spaces and that it is possible to define an inner product over $\mathcal{L}_s^2$, thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in $\mathcal{R}$ that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over $\mathcal{R}$ is consistent with the result obtained with other nonlinear generalized functions.

math.FA↗
arXiv2019

On mathematical realism and the applicability of hyperreals

We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET's argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson's infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson's techniques being applied in physics, probability, and economics, ET don't bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments `from first principles' against Robinson. In an earlier paper Easwaran endorsed real applicability of the sigma-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not sigma-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos.

math.HO↗
arXiv2019

A grid function formulation of a class of ill-posed parabolic equations

We study a nonstandard formulation of the Neumann initial value problem \begin{equation} \begin{array}{rl} u_t(x,t) = Δϕ(u(x,t)), & x \in Ω\subseteq \mathbb{R}^k, \ t \in \mathbb{R} \label{abstract}\\ u(x,0) = u_0(x), & x \in Ω. \end{array} \end{equation} with Neumann boundary conditions. The function $ϕ\in C^1(\mathbb{R})$ is assumed to be decreasing either in a bounded interval $(u^-,u^+)$, or in an unbounded interval $(u^-,+\infty)$: under this hypothesis, the aforementioned problem is ill-posed and only allows for measure-valued solutions. Moreover, such solutions are in general not unique. By using nonstandard analysis, we derive from very simple physical principles a continuous-in-time and discrete-in-space model for the ill-posed pde, and we prove that this model is well-posed. We will also prove that the solution of the nonstandard formulation is coherent with the measure-valued solutions and still retains relevant physical properties, chiefly among them an entropy condition that characterizes physically admissible solutions to the original problem. We then study the asymptotic behaviour of the nonstandard solutions. In doing so, we will give a positive answer to a conjecture by Smarrazzo on the coarsening of the solutions to the ill-posed problem under the hypothesis that $ϕ$ is decreasing in the interval $(u^-,+\infty)$.

math.AP↗
arXiv2019

Homomorphisms Between Rings with Infinitesimals and Infinitesimal Comparisons

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of "non-nonstandard" analysis. In particular, we will show that there exists a nonzero element in Henle's ring that is "too small" to be registered as nonzero in Paolo Giordano's ring, while it is seen as a nonzero infinitesimal in ${^\ast\mathbb{R}}$. This result suggests that some hyperreal infinitesimals are smaller than the nilpotent infinitesimals. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ${^{\bullet}\mathbb{R}}$.

math.RA↗
arXiv2017

Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations

We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.

math.FA↗
arXiv2014

Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.

math.HO↗
arXiv2012

Elementary numerosity and measures

In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite cardinality. We investigate the relationships between this notion and the notion of measure. The main result is that every non-atomic finitely additive measure is obtained from a suitable elementary numerosity by simply taking its ratio to a unit. In the last section we give applications to this result.

math.FA↗