In the early 18th and 19th centuries, most mathematicians thought that continuous functions were differentiable everywhere except perhaps at a few various points. In 1806, Andre Marie Ampere tried, and failed, to prove this result, which was subsequently known as Ampere’s Theorem. He also failed to provide a counterexample. Lagrange later attempted to prove the [...]
Archive for the 'Mathematics' Category
The Population Apocalypse
Published by September 30th, 2007 in Mathematics, Miscellaneous and Policy. 5 CommentsTo avoid wasting another 3 hours of my life volunteering in the Shands ER, I brought Beyond Malthus: Ninteen Dimensions of the Population Challenge to horrify me more than the bloody trauma patients being wheeled in by paramedics. Really, purchasing this book was an accident; ISIS said this was required reading for my Thermal Biology [...]
There are a few goals common to all subfields of mathematics, and they are intimately related. One is classification of mathematical objects. This is identical to the entomologist classifying species of insects, can we describe all possible objects that exist?
Here is the original. Here is the new version: “What’s your score can’t be mine”
I have always struggled with series. In a deranged email I sent to my poor calculus 2 instructor I wondered, “A series is the sum of each term in a sequence, so why does the book say s is the sum of the series? The series itself is a sum, so shouldn’t s just be [...]
Scouring various forums I discovered this transcendent mathematical quasi-love song: “But lately our relation’s not so well defined…”
The Earth is Round (p < .05)
Published by January 13th, 2007 in Mathematics and Statistics. 0 CommentsI’ve been searching for some good mathematics blogs, and I can’t seem to find but one or two. In my search, however, I did find this hilariously titled paper, The Earth is Round (p < .05) by Jacob Cohen. Naturally he cites Edward Tufte and John Tukey. See also The Standard Error of Regression (JSTOR) [...]
“The axiom gets its name not because mathematicians prefer it to other axioms.” — A. K. Dewdney, 1989
The Axiom of Choice has an interesting place in the history of mathematics, and it has produced at least one musical band as well. The description of a recent book details it nicely:
“[The Axiom of Choice,] because of [...]
And God Said, “Let there be Nonprincipal Ultrafilters”
Published by December 20th, 2006 in Mathematics. 0 CommentsApologies to all at AU for the title, but given one of Bob’s previous comments I couldn’t resist. Perhaps later we’ll discuss creation versus discovery in mathematics, but not now. So moving on….
Although we know some desired properties of largeness, we still don’t know exactly which sets are large and which aren’t. To determine this [...]
If you chose number 4, you are correct. To construct the hyperreals, we will create equivalence classes of real sequences.
In standard analysis, we can think of infinitesimals as sequences of real numbers that converge to zero and infinite numbers as divergent sequences. So the sequences
[tex] \small 1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},..[/tex] and [tex] \small \frac{1}{2},\ [...]
To see how we should construct the hyperreals, let’s consider how we can construct the reals. So what is a real number? Four common answers are….
1. A real number is an infinite decimal expansion. This isn’t a very interesting definition and it’s a bit unwieldy because you have to worry about things like non-unique decimal [...]
Modern analysis, the subfield of mathematics concerned with calculus, took an interesting turn at the end of the 19th century. By considering things like infinitely small quantities, Newton and Leibniz made the great insights that put calculus at forefront of mathematics. But worries over what exactly these infinitely small quantities were began to spread. And [...]
In one instant, he released a howl of terror. And then, the Moderate fell to his knees. Realizing this stronghold had been breached, he retreated to the citadel of freedom—the Mises Institute. And behind him came a wave of symbols, derivatives, sets, and orthodox propaganda….
Anyway, I have installed a LaTeX plugin so that AU may [...]
