Math as Myth - Preview Issue: The Story of Nautilus. Out of all of the infinite numbers in the world, there are precious few that are given their own letter from the all-too-finite Greek alphabet. The golden ratio, also known by the letter φ, or phi (usually pronounced “fie” in English), is one of those few. An irrational number that begins 1.618…, it describes an important kind of geometrical proportion—specifically, an elegant way to divide a line segment. Imagine we divide a segment (a) into a longer part (b) and a shorter part (c). If the ratio of a to b is the same as b to c, then that single ratio is golden.
A rectangle whose sides are lengths a and b is called a golden rectangle, and it’s found in the geometry of a regular pentagon and the Platonic solids, five fundamental 3-D shapes, including the cube. The golden ratio is also tightly connected with the mathematically important Fibonacci sequence: The ratios of successive numbers in the Fibonacci sequence converge to the golden ratio. In a way, this is frustrating. Mathematics under the Microscope | Atomic objects, structures and concepts of mathematics. Index of /~jeffpell/Ling406. Continuity in Topology | Mathematics Prelims. Recall a function between metric spaces and is called continuous at a point if for every there exists a such that implies . Is “near” , then will be “near” . Is continuous if it’s continuous at every point in .
If are two topological spaces, then we say that a function is continuous if for every we have ; that is, if the preimage of every open set in is an open set in . The preimage of a complement is the complement of the preimage: .The preimage of a union is the union of the preimages: .The preimage of an intersection is the intersection of the preimages: The preimage of a difference is the difference of the preimages: To see the first of these properties, simply write out the definitions for the preimages. Showing the other equalities follows the same general format of writing out the definition for the preimage and following your nose. Suppose is continuous in the metric sense. Is continuous in the topological sense as well. And let be given. Is an open subset of . . . And stay inside of . And note . . . .
Terry Tao on rigor in mathematics. (Cross-posted at NewAPPS) Fields-medalist Terence Tao (among other feats, he spotted the mistake in Nelson’s purported proof of the inconsistency of arithmetic back in 2011) has a blog post on the meaning of rigor in mathematical practice. He files this post under the heading ‘career advice’, but the post in fact touches upon some key issues in the philosophy of mathematics, such as: What is the role of intuitions for mathematical knowledge?
What is the role of formalism and rigor in mathematics? How are ‘formal’ and ‘informal’ mathematics related? While Tao’s post is not intended to be a contribution to the philosophy of mathematics as such, and while one may miss some of the depth of the discussions found in the philosophical literature and elsewhere, I find it illuminating to see how a practicing mathematician (and a brilliant one at that) conceptualizes the role of rigor in mathematical practice. Tao’s take on these matters (at least in the post) is a developmental one. Ocw.mit.edu/courses/linguistics-and-philosophy/24-244-modal-logic-fall-2009/lecture-notes/MIT24_244F09_hand02.pdf. Probability Theory — A Primer. It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post.
Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables. And so it’s about time we got to the bottom of probability theory. In this post, we will begin with a naive version of probability theory. That is, everything will be finite and framed in terms of naive set theory without the aid of measure theory. We should make a quick disclaimer before we get into the thick of things: this primer is not meant to connect probability theory to the real world. So let us begin with probability spaces and random variables.