I write about topics at the intersection of mathematics, science, philosophy, finance, and society. Posts are informal, exploratory, sometimes technical, sometimes speculative—my way of thinking out loud about ideas that excite, perplex, or motivate me. All views expressed here are my own, and should not be taken as representative of any organization or affiliation.
Latest Blog Posts
Date: November 13, 2024
Recently I had to write this proposal. I find curious the concept of a proposal. I am (theoretically) proposing myself to do something. Carefully simulating planned experiences in my head prior to their (possible) unfloding in reality. Proposal writing seems analagous to goal setting, to define a loss (objective) function. What will be minimized (maximized) during the period of the proposal?
Setting goal is a strenuous task. Firstly, you do not know where to start, so many possible targets, so many possible paths. The idea of a proposal should be to make both clear. Hopefully, by writing one, you will be able to understand (quantify) your goals. Firstly, there is the need to properly define your goals, secondly, there is the need to carefully deliniate a possible path to achieve them. At this moment, it does not need to be the shortest-most-optimal path. Such optimization processes surely will be underwent at some point. Currently, we are drawing a simple map. Once our loss (objetive) function is defined, we can start the optimization task, usually via parameter updating. There is a set goal and a path to get there, how can you reach it in the most efficient manner?
At this stage, we start exploring our map in a virtual environment, in this case, our imagination. Concepts such as gradient descent and local minima are certainly useful to have in mind. Additionally, contrafactual thinking is a great tool we have at our disposal. What would happened if I perturbed my current path, i.e. shifted slighly my parameters. One can see the paths of alternative branches of the multiverse unfolding before ones' eyes. Nothing new, old techniques adapted to modern times. This is all I have for now, hope this helps, and hope you enjoy reading my proposal.
Research
Mathematics
Machine Learning
Date: November 29, 2024
In this post I do my best to synthesize the history of financial mathematics. The title references Stephen Hawking's iconic "A Brief History of Time". The (rather short) history takes us back to the end of the XIX Century and the beginning of the XX century. For attention grabbing purposes, notable names mentioned include Albert Einstein, Henri Poincaré, Louis Bachelier, Norbert Wiener, Vito Volterra, Paul Lévy, Benoit Mandelbrot, Fisher Black, Myron Scholes, Robert Merton, Emmanuel Derman, Paul Wilmott, Edward O. Thorp. I hope you enjoy reading this short account of the history of this field. The field presents itself as the product of the efforts of countless contributors to its development and dissemination.
Financial Mathematics
History
Mathematics
Date: December 7, 2024
In this post, I explore how improved assessment methodologies can help reduce inequality and provide more accurate measures of student performance. The post discusses the merits of continuous assessment versus standardized testing, drawing on research from prominent economists and education researchers. Key topics include formative assessment practices, the challenges of different evaluation methods, and proposals for balanced approaches that combine multiple assessment types. The discussion aims to contribute to developing more equitable and effective evaluation systems in higher education.
Education
Economics
Research
Date: December 7, 2024
In this post, I explore the integration of computational tools and methods in the classroom.
Education
Technology
Ethical and Conceptual Considerations
Date: November 20, 2024
In recent years, there has been growing interest in the philosophical underpinnings of finance. This post explores how philosophical inquiry can shed light on the nature of finance, its ethical implications, and its role in society.
Philosophy
Finance
Ethics
Conceptual and Ethical Considerations
Date: November 20, 2024
In recent years, there has been growing interest in the philosophical underpinnings of finance, particularly in how mathematical concepts shape our understanding of financial systems. This post explores how philosophical inquiry into mathematics can shed light on the nature of finance, its conceptual foundations, and its ethical implications.
Philosophy
Mathematics
Finance
The Future of Human-Machine Interaction
Date: December 10, 2024
Brain-Computer Interface (BCIs) represent a groundbreaking technology that enables direct communication between the human brain and external devices. This emerging field has the potential to revolutionize various aspects of human life, from medical applications to enhancing everyday interactions with technology.
Technology
Neuroscience
AI
Date: November 27, 2024
Building upon our previous discussion of von Neumann's mathematical frameworks in finance, this post explores the ethical implications of applying these abstract mathematical structures to real-world financial systems. As financial markets become increasingly complex and automated, understanding the ethical dimensions of mathematical finance becomes crucial.
Philosophy
Mathematics
Ethics
Date: December 1, 2024
Leonhard Euler was one of the most prolific mathematicians in history, whose contributions span virtually every area of mathematics known in his time. This post highlights Euler's groundbreaking work and lasting impact on mathematics, physics, and engineering.
Mathematics
History
Date: December 2, 2024
The emergence of Large Language Models (LLMs) has opened new avenues for exploring mathematical reasoning and philosophical questions about the nature of mathematical understanding. This post examines the implications of LLMs for mathematical philosophy and the future of mathematical reasoning.
Philosophy
Machine Learning
Mathematics
Date: December 4, 2024
This post compares the perspectives of Eugene Wigner and Andrej Karpathy on the effectiveness of mathematics. Wigner's philosophical insights on the relationship between mathematics and physical reality are contrasted with Karpathy's observations on the capabilities of neural networks in capturing complex patterns.
Philosophy
Mathematics
Machine Learning
Date: December 15, 2024
Econophysics is an interdisciplinary research field that applies theories and methods originally developed by physicists to solve problems in economics. This post delves into the history, key concepts, and significant contributions of econophysics, exploring how it bridges the gap between economics and physics.
Economics
Financial Mathematics
Mathematics
On progress, stagnation and hope for the future
Date: January 22, 2025
Progress is often imagined as a linear ascent—an arrow from ignorance to enlightenment, from poverty to prosperity. Yet, as any careful observer of history or mathematics knows, the true trajectory is far more intricate, shaped by the recursive influence of prior choices. The concept of path dependence encapsulates this: past decisions, technologies, and institutions do not merely precede the present—they delimit its possibilities. When these inherited paths become "sticky," they resist even the most rational attempts at reform, and progress stalls.
Society
Economics
History
Date: November 13, 2024
Historical context with an emphasis in Portugal and Spain and their inefficiency caused by the Court System. Innovation as a driver of successful societies. GDP per capita as a measure of success. Darwinism of Institutions.
Society
Economics
History
Date: January 29, 2025
Self-reference lies at the heart of intelligence. It allows systems—biological or artificial—to construct internal models of themselves, predict their own behavior, and learn from self-generated representations. Yet it also defines the ultimate boundary of formal reasoning: the point beyond which a system cannot consistently describe itself without paradox. From Gödel’s incompleteness to Turing’s halting problem and modern self-modeling AI, self-reference has emerged as both the mechanism of understanding and the source of undecidability. This article examines how reflexivity functions as the structural essence of intelligence, and why every attempt at self-understanding—whether logical, computational, or cognitive—confronts the same recursive horizon.
Philosophy
Mathematics
AI
Logic
Date: October 26, 2025
This post explores the intersection of Kolmogorov complexity and fractal geometry, introducing the concept of algorithmic (effective) dimension. We discuss how the shortest description of an object (from algorithmic information theory) connects with how detail scales with resolution (fractal geometry), and how these perspectives unify in understanding the geometry of information.
Mathematics
Information Theory
Fractal Geometry