Spaces upon spaces – topology and slum conditions

I didn’t know anything about topology before I entered graduate school, but I continue to see it as one of the more provocative specialties in mathematics and an important transition of thought. Most definitions of the subject describe it as the study of the properties of objects that are preserved through deformations like stretching and […]

Prejudice in an abstract world

I was struck today by the title of an article in Science News that read, Before his early death, Riemann freed geometry from Euclidean prejudices. The piece, by science writer Tom Siegfried, was no doubt inspired by the recent claim from award-winning mathematician Michael Atiyah that he has proved the long standing Riemann hypothesis, one […]

To be or not to be abstract

I’m not completely sure I understand where my desire to grasp the value of abstractions is taking me, but as I think about mathematics, and more recent trends in the sciences, I keep wanting to get further and further behind what our symbolic reasoning is actually doing, and how it’s doing it. I have this […]

Abstractions: What’s happening with them?

We all generally know the meaning of abstraction. We all have some opinion, for example, about the value of abstract painting. And I’ve heard from many that mathematics is too abstract to be understood or even interesting. (But I must admit, it is exactly this about mathematics that keeps me so captivated). An abstraction is […]

The fluency of geometry

My thoughts started jumping around today, trying to land on what it was that I found so fascinating about a recent article in Quanta Magazine. This is one of the statements that got me going:

…Numbers emerging from one kind of geometric world matched exactly with very different kinds of numbers from a very different […]

Mathematical hybrids and the like

My attention was just recently brought to the work of philosopher and poet Emily Grosholz. It’s rare to find an individual so steeped in the ways of poetry and mathematics, and the desire to explore how and what they express about us. What I would like to consider here, in this particular post, is really […]

I spy the confluence of mathematics, psychology, and physics

I find the relationship between mathematics and vision fascinating. Even within mathematics itself, seeing how the geometric expression of ideas can clarify or further develop countless mathematical thoughts is always worth noting – like the graphs of functions, or the projections of figures. I’ve written before about the relationship between the brain’s visual processes and […]

Spaces

I read about the sad passing of Maryam Mirzakhani in July, and the extraordinary trajectory of her career in mathematics. But I did not know much about what she was actually doing. A recent post in Quanta Magazine, with the title: Why Mathematicians Like to Classify Things, caught my attention because the title suggested that […]

Mathematical behavior without a brain?

I have made the argument on more than one occasion that a refreshed look at mathematics may help illuminate the relationship between our experience of the physical and our experience of the thoughtful. Mathematics is a discipline characterized by complex relations among abstract things but, as has been explored from many directions, the action of […]

The Mind of Arithmetic

I like finding what I believe are productive generalizations, or some sameness across apparently diverse events. Often the subject of Mathematics Rising are things like the observation that we seem to search for words in patterns that resemble the trajectories of a rat foraging for food, or the possibility that natural language can be described […]