Category Archives: Math

Oppenheimer: on Math

Source: InfoProc, Jan 2017

Oppenheimer: Mathematics is “an immense enlargement of language, an ability to talk about things which in words would be simply inaccessible.”

Quotes – Ramanujan

Source: WikiQuote, date indeterminate

Paul Erdős has passed on to us Hardy‘s personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100.

  • Bruce C. Berndt in Ramanujan’s Notebooks : Part I (1994), “Introduction”, p. 14

The formulae (1.10) – (1.13) are on a different level and obviously both difficult and deep… (1.10) – (1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.

Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him.

  • Michio Kaku, Hyperspace : A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (1995), p. 172

The great advances in mathematics have not been made by logic but by creative imagination. The title of mathematician can scarcely be denied to Ramanajan who hardly gave any proofs of the many theorems which he enumerated.

GH Hardy (via Paul Erdos) on Ramanujan

Source: Famous Scientists, date indeterminate

He was told an interesting story by the great Hungarian mathematician Paul Erdős about something G. H. Hardy had once said to him:

“Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100.”

Transforming Math Equations into 3D Figures

Source: Quanta magazine, Jan 2017

Mathematicians are not so different from naturalists. Rather than studying organisms, they study equations and shapes using their own techniques. They twist and stretch mathematical objects, translate them into new mathematical languages, and apply them to new problems. As they find new ways to look at familiar things, the possibilities for insight multiply.

a new idea from two mathematicians: Laura DeMarco, a professor at Northwestern University, and Kathryn Lindsey, a postdoctoral fellow at the University of Chicago. They begin with a plain old polynomial equation, the kind grudgingly familiar to any high school math student: f(x) = x2 – 1. Instead of graphing it or finding its roots, they take the unprecedented step of transforming it into a 3-D object.

With polynomials, “everything is defined in the two-dimensional plane,” Lindsey said. “There isn’t a natural place a third dimension would come into it until you start thinking about these shapes Laura and I are building.”

a promising new method of inquiry: Using the shapes built from polynomial equations, they hope to come to understand more about the underlying equations — which is what mathematicians really care about.

“These are fascinating and beautiful things that arise very naturally in our subject and should be understood!” DeMarco said by email, referring to the shapes.

in 2010 William Thurston, the late Cornell University mathematician and Fields Medal winner, heard about the shapes from McMullen. Thurston suspected that it might be possible to take flat shapes computed from polynomials and bend them to create 3-D objects. To explore this idea, he and Lindsey, who was then a graduate student at Cornell, constructed the 3-D objects from construction paper, tape and a precision cutting device that Thurston had on hand from an earlier project. The result wouldn’t have been out of place at an elementary school arts and crafts fair, and Lindsey admits she was kind of mystified by the whole thing.

“I never understood why we were doing this, what the point was and what was going on in his mind that made him think this was really important,” said Lindsey. “Then unfortunately when he died, I couldn’t ask him anymore. There was this brilliant guy who suggested something and said he thought it was an important, neat thing, so it’s natural to wonder ‘What is it? What’s going on here?’”

DeMarco and Lindsey’s work is heavily influenced by the mid 20th-century mathematician Aleksandr Aleksandrov. Aleksandrov established that there is only one unique way of folding a given polygon to get a 3-D object. He lamented that it seemed impossible to mathematically calculate the correct folding lines. Today, the best strategy is often to make a best guess about where to fold the polygon — and then to get out scissors and tape to see if the estimate is right.

“I would even characterize it as being sort of playful at this stage,” McMullen said, adding, “In a way that’s how some of the best mathematical research proceeds — you don’t know what something is going to be good for, but it seems to be a feature of the mathematical landscape.”

A Geometry of Thought

Source: InfoProc blogspot, Dec 2016

I can share what I have been thinking about lately. In Thought vectors and the dimensionality of the space of concepts (a post from last week) I discussed the dimensionality of the space of concepts (primitives) used in human language (or equivalently, in human thought). There are various lines of reasoning that lead to the conclusion that this space has only ~1000 dimensions, and has some qualities similar to an actual vector space. 
an automated method to extract an abstract representation of human thought from samples of ordinary language. This abstract representation will allow machines to improve dramatically in their ability to process language, dealing appropriately with semantics (i.e., meaning), which is represented geometrically.
an automated method to extract an abstract representation of human thought from samples of ordinary language.

Gowers: Ideas for UK math teaching

Source: Gowers website, Jun 2012

What I emphatically would notlike to see is teachers learning “the right answer” and giving a mini-lecture about it to their classes. Instead, the entire discussion should be far more Socratic.

The idea is that the teacher would go into a discussion about a question like this with a good grasp of the issues involved, but would begin by simply asking the question. An initial danger is that nobody would have anything to say, but one way of guarding against that is to discuss questions that people are likely to care about.

For example, the question above about whether girls are better than boys at a certain subject is far more likely to encourage people to think critically about statistics than a mathematically equivalent question about a less contentious topic. If the discussion stalled, the teacher’s job would be to give it a little nudge in the right direction.

The main point is one I’ve basically made already: the discussions should start from the real-life problem rather than starting from the mathematics. Pupils should not feel that the question is an excuse to force some mathematics on them: they should be interested in the question and should feel the need for the mathematics, the need arising because one can give much better answers if one models the situation mathematically and analyses the model.

Feynman: Mathematics is Finding Patterns

Source: The Physics Teacher, 1968

If I were giving a talk on “what is mathematics,” I would already have answered you. Mathematics is looking for patterns.