Blog
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In the previous post of this thread, I explored the nature of forces. In this post, I will go into more detail on how forces act on physical objects. The central law is Newton’s third law. Action-Reaction Pairs (Newton’s Third Law) Newton’s third law provides a profound insight into how forces actually arise: every force… Continue reading.
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In our previous discussions, we explored Newton’s laws of motion and saw how reference frames shape our description of motion. At the heart of Newtonian mechanics is the concept of a force—a physical influence capable of changing an object’s state of motion, causing acceleration. Forces provide the fundamental way objects interact with one another. Whenever… Continue reading.
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This post continues the exploration of predicate logic, highlighting its expressiveness compared to propositional logic, discussing its limitations, and outlining applications in various fields. Advanced Expressiveness Compared to Propositional Logic Propositional logic treats whole sentences as indivisible. Predicate logic can talk about individuals and relations between them, and it can express generality and existence. Here… Continue reading.
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Having established both syntax and semantics, we now delve into the essential methods used to formally prove statements in predicate logic. Specifically, we’ll explore Natural Deduction and Hilbert-style systems, illustrating their application to real mathematical reasoning. Natural Deduction in Predicate Logic Natural deduction is a formal proof system that extends naturally from propositional logic, incorporating… Continue reading.
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In the previous posts, we’ve discussed the syntax of predicate logic, outlining how terms, predicates, and formulas are formed. Now, we’ll explore semantics, explaining how meaning is formally assigned to these formulas. Structures and Interpretations The meaning of formulas in predicate logic is given by structures (or interpretations). Intuitively, a structure assigns concrete meanings to… Continue reading.
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In my previous post, I introduced variables and explained how C++ stores and manages data using fundamental data types. Now, I will delve deeper into how memory works in C++ and introduce two powerful features: pointers and references. Understanding memory management is crucial for becoming proficient in C++. It will give you greater control over… Continue reading.
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In numerical computations, errors can propagate through calculations, potentially leading to significant inaccuracies in results. Understanding how errors propagate and how the conditioning of a problem affects numerical stability is crucial for designing robust numerical algorithms. In this post, I will discuss error propagation and the concept of conditioning in numerical problems. Error Propagation Errors… Continue reading.
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Building on our understanding of stochastic processes and Brownian motion, we now delve deeper into the mathematical framework essential for modeling financial systems—Itô calculus and stochastic differential equations (SDEs). These tools allow us to rigorously handle randomness in continuous-time finance models. Stochastic Differential Equations Explained A stochastic differential equation (SDE) describes how a stochastic process… Continue reading.
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In the previous post, we saw why propositional logic is not sufficient to express general mathematical statements, and why predicate logic is required.In this post, we’ll explore the formal syntax of predicate logic, detailing how statements are constructed using terms, predicates, quantifiers, and formulas. Terms In predicate logic, terms represent objects within a given domain.… Continue reading.
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Why Predicate Logic? In our journey through formal mathematics, we’ve explored propositional logic—a powerful yet limited tool. Propositional logic allows us to reason about the truth of statements built from simpler ones using logical connectives. However, it falls short when we need to express statements about generalizations, properties, or specific relationships involving objects. Predicate logic… Continue reading.
