This is the introduction I wrote for the multi-authored book "From Riemann to differential geometry and relativity", edited by L. Ji, A. Papadopoulos and S. Yamada (Berlin, Springer verlag, 2017). The book consists of twenty chapters, written by various authors. This introduction, besides giving the information on the …
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48 results for “philosophy of mathematics”
Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorummath.GT
This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.
problem Understanding the historical context and philosophical implications of angles and solid angles.
method Historical review and analysis of mathematical and philosophical works.
result Questions raised by Euler about angles and solid angles are timeless and relevant to modern mathematics.
Curvature of surfaces in euclidean 3-space: philosophical analysis of Gauss' Theorema Egregiummath.HO
This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically invariant character of curvature, struggled with elimination computations in order…
This paper argues that the fundamental principle of contemporary financial economics is balanced reciprocity, not the principle of utility maximisation that is important in economics more generally. The argument is developed by analysing the mathematical Fundamental Theory of Asset Pricing with reference to the emergen…
Riemann's mathematical papers contain many ideas that arise from physics, and some of them are motivated by problems from physics. In fact, it is not easy to separate Riemann's ideas in mathematics from those in physics. Furthermore, Riemann's philosophical ideas are often in the background of his work on science. The …
Model credit ratings using economic states with Markov chains.
problem Credit rating migration influenced by economic state changes.
method Developed a Markov chain model for credit ratings conditional on economic states.
result Derived asymptotic behavior of the rating process using Markov theory.
This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. …
Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model …
The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …
The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within well-adapted models…
Bayesian framework explains diverse explanatory values.
problem Understanding and predicting human preferences for explanations.
method Developed a Bayesian account to integrate various explanatory values.
result Core values from psychology, statistics, and philosophy emerge from a common framework.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an -type -cluster structure.
Machine learning's data-centric philosophy conflicts with natural sciences' standards.
problem Conflict between machine learning's ontology and epistemology and natural sciences' practices.
method Identifying and analyzing contexts where ML can be beneficial or harmful in natural sciences.
result ML can enhance trustworthiness in causal inference but introduces biases in emulation and labeling.
Developable ruled surfaces generated by curvature axes of curves.
problem Creating simple and understandable ruled surfaces for practical design.
method Investigating a straightforward method to generate developable ruled surfaces using curvature axes of curves.
result Developable ruled surfaces are generated by the curvature axes of curves, and these surfaces are developable.
This book presents a methodology and philosophy of empirical science based on large scale lossless data compression. In this view a theory is scientific if it can be used to build a data compression program, and it is valuable if it can compress a standard benchmark database to a small size, taking into account the len…
Paper connects fair machine learning to political philosophy, highlighting flaws in ideal approaches.
problem Lack of natural formulation for social desiderata in machine learning.
method Proposes metrics and algorithms to satisfy subsets of fairness parities, trading off against utility.
result Misguided fair machine learning algorithms reflect broader flaws in ideal methodological approaches.
Eliashberg simplifies singularities in geometry.
problem Complex singularities in geometric structures.
method Philosophy of the h-principle and simplification techniques.
result Simplified understanding of singularities in geometry.
One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agent-based viewpoint in finance goes back at least to the work of Garman (1…
Paper proposes a new AI design philosophy for distributed learning.
problem Designing AI systems that mimic human cognitive capabilities.
method Democratized learning (Dem-AI) with self-organized hierarchical agents.
result Self-organized learning systems can perform complex tasks more efficiently.
A new clustering algorithm inspired by Wittgenstein's philosophy.
problem Clustering data without assuming predefined cluster shapes or sizes.
method Wittgenstein's family resemblance concept applied to machine learning.
result WFR clustering algorithm effectively identifies clusters in data.
Quant 4.0 uses AI to automate, explain, and incorporate knowledge in investment.
problem Limitations of deep learning in quant investment.
method Automated AI, Explainable AI, Knowledge-driven AI.
result Improves investment decision-making through automation, interpretability, and prior knowledge integration.
Proposes a new trust framework for AI models to maximize utility.
problem Concerns over bias and discrimination in predictive models.
method Introduces a novel trust framework inspired by philosophy, focusing on maximizing Bayes utility.
result Properly-ranked models are inherently -trustworthy.
The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working on…
Formulates approach for guiding explanation types based on user specifications.
problem Creating explainable AI components from user-defined specifications.
method Develops a method for generating explanations based on user-defined specifications.
result Demonstrates feasibility of user-defined explanations for complex models like Bayesian networks and graph neural networks.
We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipsc…
Torsion Energygr-qc
In the present work, torsion energy is defined. Its law of conservation is given. It is shown that this type of energy gives rise to a repulsive force which can be used to interpret supernovae type Ia observations, and consequently the accelerating expansion of the Universe. This interpretation is a pure geometric one …
New method reduces indirect discrimination in insurance risk models.
problem Indirect discrimination in insurance risk models using machine learning.
method Mathematical concepts of linear algebra to reduce indirect discrimination.
result Demonstrated promising performance in a concrete case of risk selection in life insurance.
Can an interdisciplinary field contribute to one of the parent disciplines from which it emerged?physics.soc-ph
In the light of contemporary discussions of inter and transdisciplinarity, this paper approaches econophysics and sociophysics to seek a response to the question -- whether these interdisciplinary fields could contribute to physics and economics. Drawing upon the literature on history and philosophy of science, the pap…
Weyl's 1918 geometry proposal revisited in modern physics.
problem Revisiting Weyl's 1918 geometry proposal in modern physics.
method Reconsideration of Weyl's scale gauge in high energy physics and gravitation theory.
result Weyl geometry has regained interest in modern physics, particularly in particle physics and cosmology.
The Frame Problem (FP) is a puzzle in philosophy of mind and epistemology, articulated by the Stanford Encyclopedia of Philosophy as follows: "How do we account for our apparent ability to make decisions on the basis only of what is relevant to an ongoing situation without having explicitly to consider all that is not …
Floer Field Philosophymath.SG
Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension…
Paper resolves apparent conflict between group and individual fairness.
problem Apparent conflict between group and individual fairness in machine learning.
method Theoretical discussions from fair machine learning, political, and legal philosophy.
result Individual and group fairness are not fundamentally in conflict.
Pylearn2 is a machine learning research library. This does not just mean that it is a collection of machine learning algorithms that share a common API; it means that it has been designed for flexibility and extensibility in order to facilitate research projects that involve new or unusual use cases. In this paper we g…
1-domination of knotsmath.AT
We say that a knot in the -sphere {\it -dominates} another if there is a proper degree 1 map between their exteriors, and write . When but we write . One expects in the latter eventuality that is more {\it complicated}. In t…
No free lunch theorems show all algorithms perform equally under uniform distribution.
problem Analyzing scenarios involving non-uniform distributions and comparing algorithms.
method No Free Lunch theorems applied to analyze and compare algorithms without distribution assumptions.
result Anti-cross-validation performs as well as cross-validation under non-uniform distributions.
This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy (2009) and related to the recent contribution Carmona and Nadtochiy (2012) by the same authors, the key parametrisation of our app…
A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Ratesmath.AP
For gravitational collapse, we observe a correspondence between region close to past null infinity and region close to central singularity. In line with this philosophy, we construct a new ansatz, with which we first present a 40-page self-contained proof of trapped surface formation in a far-from-center region. A syst…
The Epicurean Philosophy is commonly thought as simplistic and hedonistic. Here I discuss how this is a misconception and explore its link to Reinforcement Learning. Based on the letters of Epicurus, I construct an objective function for hedonism which turns out to be equivalent of the Reinforcement Learning objective …
New algorithms learn in complex decision-making problems with smooth transitions.
problem Learning in complex decision-making problems with smooth transitions.
method UCB and PSRL philosophies applied to episodic Markov decision processes with kernel approximation.
result Low regret learning achieved in continuous state and action spaces.
Machine learning disciplines shift values, not just model types.
problem Values influence machine learning discipline and model types.
method Philosophy of science lens, conceptual framework analysis.
result Disciplinary shifts encode social and political values.
Distiller simplifies DNN compression research with a Python package.
problem Efficiently compressing deep neural networks.
method Open-source Python package with DNN compression algorithms.
result Facilitates new research and learning tasks in DNN compression.
LAGE is a systematic framework developed in Java. The motivation of LAGE is to provide a scalable and parallel solution to reconstruct Gene Regulatory Networks (GRNs) from continuous gene expression data for very large amount of genes. The basic idea of our framework is motivated by the philosophy of divideand-conquer.…
New approach for large-scale distributed learning systems that improve generalization performance.
problem Transitioning from centralized to distributed AI systems for complex learning tasks.
method Self-organizing hierarchical structuring mechanism based on agglomerative clustering, hierarchical generalization, and personalized learning.
result Demonstrates better generalization performance compared to conventional federated learning algorithms.
We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of …
Market Crashes as Critical Phenomena? Explanation, Idealization, and Universality in Econophysicsphysics.hist-ph
We study the Johansen-Ledoit-Sornette (JLS) model of financial market crashes (Johansen, Ledoit, and Sornette [2000] "Crashes as Critical Points." Int. J. Theor. Appl. Finan. 3(2) 219-255). On our view, the JLS model is a curious case from the perspective of the recent philosophy of science literature, as it is natural…
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of c…
In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using Cerf Theory, developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Waldhausen's uniqueness of Heegaard splittings of S^3. In this note we use Cerf Theory …
The paper refines Mather-Thurston theorems for flat connections in manifolds.
problem Understanding flat connections in manifolds and their obstructions.
method Refined cobordism and control of transition functions for larger structure groups.
result Controlled cobordisms and detailed transition functions for flat connections.