There are countless ways to explore what may be called the two faces of mathematics – algebra and geometry. Modern mathematical systems have their roots in both algebraic and geometric thinking. Like the organs of the body which are built on the redirected sameness of cells, algebra and geometry live in all manner of relationship […]
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My last post caused me to survey some things related to Bayesian statistics as they relate to mathematics and cognition. First, I want to say that despite the fact that I have been looking more closely at 19th century developments in mathematics, I didn’t know until today that Laplace, in 1814, described a system of […] Some of George Berkeley’s fame comes from his vehement critique of Newton’s calculus. His criticism was harsh and inspired a number of responses from contemporaries who accepted the vanishing quantities Newton used to formulate his notion of fluxions or, in modern terms, his understanding of instantaneous rates of change. The discussion that followed Berkeley’s 1734 […] I have spent some time pointing to milestones in the history of modern mathematics where a conceptual shift produces provocative new thought – as when Riemann gave a new foundation to geometry, or when Cantor brought precision to the notion of countability. Modern mathematics, partnered with physics, increasingly refines what the human mind can perceive. […] I had the opportunity to attend a talk given by Frank Wilczek, Nobel laureate in physics and author of the book The Lightness of Being. During the Q and A after the talk he was asked if our aesthetic judgment of symmetry could be said to prejudice scientific inquiry. Wilczek first pointed to the rich […] I don’t think it’s actually possible to answer the question in the title of this post, but I still believe it’s worth asking. We’ve thought of things ‘hidden under a microscope,’ or obscured by great distances, but in mathematics when something is hidden, it’s because we haven’t been able to imagine it yet. And when […] I don’t have a lot of time to write this week and next but I felt a little surge of thoughts gather when I read about the implications of Einstein’s Second Law (m = E/c^2) which is a simple rearrangement of the now famous E = mc^2. It was in The Lightness of Being by […] It’s easy to neglect the detail of one person’s, now historic, philosophical discussion of math and science. But there is a moment, in Henri Poincare’s well known text Science and Hypothesis, that I would like to shine a light on today. The first English translation of the book was published in 1905. Chapter IV is […] In the first half of the twentieth century, physicists and mathematicians began to raise questions about what they could say about what they were actually doing. The ‘truth’ of things was beginning to elude the seekers of that truth. Both the validity of mathematical ideas and the objectivity of physics came under scrutiny. Questions about […] I’ve become a bit preoccupied recently with the world of early 20th century mathematicians, partly because of a book I’m working on, but also because of how late 19th and early 20th century thinking largely defines the mathematics students learn today. In this light I found a book of selected writings of Hermann Weyl, a […] |
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