Mathematical logic

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In “The Nature of Mathematical Reasoning,” Henri Poincaré argues that mathematical facts are synthetic and a priori, because we can see the principle of induction is true by non-analytic methods. Exponents of logicism, such as Carl Hempel, would disagree with Poincaré because they would see induction as following analytically from the definition of natural numbers. However, this response fails to fully address the argument, because Hempel’s logical approach cannot capture human intuition about…

Descartes asserts that aspects of mathematics and geometry—like the triangle postulate—exist innately in his mind, yet he also “clearly and distinctly perceive[s]” the natures of the properties of such concepts (Med V.68). Descartes’s mathematical certainty does not stem from physically viewing a triangle with his eyes or feeling a triangle’s three sides with his hands, instead, Descartes clearly and distinctly “perceives” the innate and unchangeable properties of a triangle—its three sidedness…

As Christians, it is of utmost importance that we thoughtfully study how we know what we know. In order to be able to give proper defense of our faith, we must be able to articulate why we believe what we believe, rather than absentmindedly resigning to our Christian worldview. This articulation of our beliefs is essential within the bounds of our respective fields as we often fail to recognize the underlying philosophical natures in the disciplines to which we devote the most time. Therefore, I…

Mathematics, to me, is not just a tool, nor is it only a language for reading our universe. To me, mathematics is an art: the art of logic; and like any other form of art, it is beautiful. This is because the mathematical universe is perfect: here every piece of argument, every theorem is a constellation of reasoning all built upon the strict axioms of the mathematical universe. The proper way of exploring this beauty is problem solving. By problem solving, I do not mean simply trying to fit a…

There is a clear distinction between rationalism and empiricism–two theories of epistemology, which focuses on the theory knowledge through a philosophical lens. Rationalism centers around the ability to use knowledge with reasoning and that knowledge comes from innate ideas. Others believe that our senses could actually lead to true knowledge, as opposed to relying on innate ideas and reasoning. Many criticize rationalism and whether innate knowledge exists or is even pertinent. In this paper,…

order for deeper understanding to take place. I believe, based on my prior experiences and research, that active learning allows students of all learning styles to gain a deep mathematical understanding of content. Active learning also offers opportunities for students to explore and utilize the eight Standards for Mathematical Practices that are listed in the Common Core State Standards. To accommodate for the various learning styles of students, I believe that a variety of assessment…

to back up reasoning, it had as many weaknesses as strengths. It’s why reason as a way of knowing is not reliable as we could think. In this essay, I will to put in evidence the strengths and the weaknesses of reason through deductive and inductive logic, ethic, mathematic, medicine and finally religious aspects. The first strength of reasoning is that it seems to give certainty to what we say. It gives us certainty thanks to a correct logical reasoning. A correct reasoning involves proves,…

illustrate how the mind works the concepts introduced previously. The three paradoxes are: Berry paradox, Richard’s paradox, and the paradox of the Liar. All three paradoxes explore the limits of human languages and how that affects our understanding of mathematical concepts. The Berry paradox is particularly concerned with numbers and their descriptions; there are numbers that are cannot even be named for example. I liked how the author described the relationship of the Berry paradox to…

American diet is life threatening. The health risks that come with eating over processed food with too much sugar are costly and dangerous. His solution is to tax unhealthy foods. Although a simple, money-oriented solution is often criticized, his logic and reasoning is surprisingly convincing. Asking to increase the cost of everyone’s favorite foods like doughnuts, candy, and soda, is a hard argument to…