About the project / FAQ

What do research mathematicians do? What does math research feel like? What are the different ways that math researchers use their minds?

Mathematics is fundamentally about ideas.  Ideas have a rigorous objective dimension, but they also have an experiential dimension.  Mathematicians are really good at talking about the rigorous objective dimension of their ideas.  It’s much harder to talk about the experience of the ideas, the experience of doing mathematics.  This blog is my attempt to do a better job at describing and analyzing the mathematical experience, more specifically the research experience.

I need your help!  If you read some of the posts and are interested in posting your own flavor or season, go to the Post Your Own page.

I’m focusing on two general classes of experience.  A “flavor” is a mathematical experience that you might have for a few hours or a day.  A “season” is a mathematical experience that spans weeks or months (or years).  In describing these experiences, I’m interested in cognitive, meta-cognitive, emotional, and logistical data.  More specifically, each flavor and season description will be framed by certain questions.

Questions for each flavor:

  1. What’s going on mathematically?
  2. What is the emotional and logistical context?
  3. What thoughts are there?
  4. What quality of awareness?
  5. What emotions?
  6. What does it resolve to, after how much time?
  7. How frequent is this flavor?
  8. What are good/bad ways to change or follow it up?

Questions for each season:

  1. What mathematical activities characterize it?  What level of rigor?
  2. What relevant interactions with other mathematicians?
  3. How does it feel, what is the mood?
  4. What state of mind? stable vs. chaotic? focused vs. dispersed?
  5. What type of self-reflection during the experience, and does it help?
  6. An everyday metaphor for the experience?
  7. An example of a good and bad day?
  8. What did you do when you were stuck?
  9. When and why did it end?

My goal is to attempt to catalogue recurring experiences, with the hope that at some point this project will shift from a personal one to a collective one.  My main interest is not in talking only about myself, but in finding common threads among the experiences of different mathematicians – things we all go through.  The key word is “intersubjective” –  subjective experiences take on a degree of objectivity when we talk about them, and map out the commonalities that transcend the individual.

Aside: {There is an important and fascinating aspect of the personal mathematical experience that I’m not addressing in this blog.  Specific mathematical concepts invoke specific mental imagery, and because of our shared intellectual upbringing, many mathematicians find agreement among these internal images and metaphors.  For example, I think most people understand the negative numbers via a visual image of numbers on a line.  “Rigorously,” the negative numbers are independent of this image.  But as an objective-subjective external-internal phenomenon, experienced within separate minds but couched in a shared culture, the idea of negative numbers is not independent of the mental imagery it invokes.  If you don’t understand the visual image of the line, you don’t understand negative numbers.  This fascinating subject is hardly appreciated or acknowledged as much as it should be, but for whatever reason I don’t feel like filling this blog with such examples.  If you’re interested, I recommend: Where Mathematics Comes From by George Lakoff and Raphael Nunez; The Psychology of Invention in the Mathematical Field, by Jacques Hadamard, or, for example, this Math Overflow post started by Bill Thurston.}

If you pin them down and force them to self-reflect, in my experience most mathematicians rather enjoy the challenge of meta-analyzing their thought processes.  However, there are a few standard criticisms of such contemplations and discussions, which I will now list and respond to.

Criticism A: These sorts of discussions are too personal, subjective, and imprecise.

Response: I’m going to try to describe everything in as much detail as possible. And not just emotions, but concrete mental images and mental textures. So things will be concrete, and you can read them and either agree or disagree. And you can write your own. But mind you, these ideas will never have the same objectivity and precision as math. Math is singular in its precision, and that precision has allowed us to build such intricate structures and theories. But you don’t need such extreme precision to say something interesting. Evolution is a very interesting idea, but the mechanisms involved, the different manifestations, all the exceptions to the rules, all the varying data – you’ll never pull everything together as definitively as, for example, the theory of finitely generated abelian groups. That doesn’t mean evolution is nonsense though.

Criticism B: These conversations never go anywhere, so they’re not worth having.

Response: They never get far, because we can’t define our terms as rigorously as mathematical terms, and that frustrated us. I’m trying to establish a database of flavors, a somewhat precise vocabulary that we can build on and refer to. Even if we fail to find any overlap in our mathematical experiences, I think the self-reflection, precise or vague, necessary to reach that conclusion is valuable. Even if I fail to bring any precision to the venture, the contemplation makes us better mathematicians.

Criticism C: These conversations are interesting, but they’re not math.

Response: In one sense, I don’t really care what we do or don’t call math.  In another sense, you’re right – this is more like math psychology.  In a third sense,  I think that math is that which mathematicians do, and ideas are interior, as well as exterior, phenomena.

This criticism seems to take rigor as the defining characteristic of mathematics, and I disagree with this. I think math is ideas, and ideas can have a rigorous dimension, as well as other dimensions. I wrote a book, “My Name is Not Susan: A Love Story Between Mathematics and Non-Mathematics,” that in the final chapter traces the historical origin of this criticism, and suggests a relaxation of our notion of objectivity and of our insistence on rigor. You can download the book for free here.  That chapter is inspired by William Byers’ How Mathematicians Think; he does a more thorough job than I do.

To start with, I will be posting flavors and seasons from my own mathematical life.  But I hope that I can get others involved in this project.  If you’ve read this far and are interested, go to the Post your own page.


1. Who do you think you are?

I’m Luke Wolcott, a recent mathematics PhD, now on a postdoc at IST in Lisbon, with Dan Christensen of the University of Western Ontario.  My PhD was at the University of Washington, in Seattle, working with John Palmieri, in algebraic topology.  The three-sentence cocktail-party explanation of my research: “Take the collection of all possible spaces, of all possible dimensions – positive and negative – and try to understand the global structure of this whole collection.  This is really hard, so it’s a good idea to simplify by zooming out and making things a little fuzzy.  There’s two ways to zoom out, and I study the differences between them.”

My website – with more essays on math, and at least one math dance video – is at www.forthelukeofmath.com.

2. Why are you doing this?

I’ve found that increased self-awareness of my mathematical process makes me a better mathematician.  I think that the better we understand how we do math, the better we’ll understand how we do our best math, and the more beautiful math we’ll create.  So this blog is a challenge for me, and an invitation for you, to become more aware of how we do math.

If I’m right, and there are overlaps in my experience of math and yours, then maybe you and I can gain from the sharing, commiserating, troubleshooting, and appreciating of those common experiences.

I also believe there is an important intersubjective reality to our mathematical concepts, and the experience of discovering/ creating/ teaching/ learning/ explaining those concepts.  Understanding intersubjectivity – the fuzzy internal/external realm between subjective and objective, is fascinating to me, and seems like the next crucial step if our culture is going to attempt to reconcile science with consciousness, or science with spirituality, or body with mind.

3. How, exactly, is self-reflection going to make me a better mathematician?

By tweaking how you do math – how and when you do what kind of math – you can become more efficient, have more insights, and enjoy it more.  You’ll spend more time, I claim, doing the right math at the right time.  How will this blog help?  Read about the different flavors and seasons, and reflect on them – have you experienced that? have you tried to do math that way? how was it similar and different for you? Maybe you can try new things, or, with more perspective, embrace the challenges?

4. Why are you focusing on research and not teaching – that’s a part of the mathematical experience, isn’t it?

It’s common for anyone who expresses an interest in the human side of mathematics to be pushed out of the “research” camp and into the “teaching” camp.  This selection process has left the researchers with too little self-reflection on the interior dimensions of their practice, in my opinion.  On the other hand, many great minds have made great progress in understanding how best to teach calculus, or the common hurdles all students must jump over to master fractions, etc.  A good teacher never stops thinking and talking about the internal experience of math ideas; a good researcher may think about it, but won’t likely talk about it.  So I’ve made the decision to focus on the researchers.

5. Will this help me do my homework?

As mentioned above, I’m going to avoid discussing the microsecond-scale imagery and metaphors that mathematicians use when they think, although it’s fascinating and I’m sure there is intersubjectivity there.  I’m also going to avoid discussing problem-solving techniques, e.g. “When you’re stuck, try changing your perspective.”  I think an awareness of these universal tools does make us better at doing math, but the subject is too big to treat here.  If you’re interested, I recommend the classic How To Solve It, by George Polya, or Mathematical Problem Solving, by Alan Schoenfeld, or even better, musician Brian Eno’s Oblique Strategies.  (There’s a good chance some problem-solving discussion will come up, but I don’t plan to focus on it.)

6. How is this different from other math blogs?

Well, it’s not really any different, I guess.  Although I don’t think of self-reflection and thinking about math as tangential, secondary, or in any way less important than thinking in math.


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