This post is the first on a meditation on the nature of mathematics as I see it. I have been thinking about this for some time, and my thoughts were again stimulated by a March 2014 article I read in Slate by Brian Palmer that attempted a popularized explanation of the mathematical concepts associated with Zeno’s Paradox. It was a laudable effort that I applaud. So it is a bit churlish of me to critique it, but I felt its misconceptions got at the heart of some fundamental ideas about mathematics that I wanted to clarify.
The key idea exemplified in this article is the role “making it up” plays in math. That is, the general impression seems to be that math is dealing with things as they actually are if we can just be brought to see it. Whereas the idea that mathematicians make things up or define things is given little credence. For example, 0 x 2 “is” 0 doesn’t make any sense if you arrive at multiplication inductively from the intuitive idea of its being repeated addition. That is, 2 x 0 = 0 + 0 = 0 makes sense, but 0 x 2 = 0 does not. So mathematicians just say let’s define 0 x 2 = 0. If we do, it will be consistent with the other rules we have abstracted from the repeated addition idea, such as the commutative and distributive rules – that is, nothing breaks. (Try defining 0 x 2 to be any other number than 0 and see what breaks.) To put it another way, the reason we want to have 0 x 2 = 0 is for a different reason than we originally thought was meant by multiplication. We have extended the original idea into new territory. A similar thing happens with the advent of negative numbers. This is a very sophisticated idea and a challenge to present at an elementary stage.
In Part I, I will first present the article, heavily annotated with my critique. Then in Part II I will try to explain in more depth the admittedly philosophical concepts I am trying to get at. See Meditation on “Is” in Mathematics I – Zeno’s Paradox.
James Tanton has provided further elaborations on the polygons and the sum of perpendicular distances from interior points. Again I approached the solutions with a mix of areas and vectors. It is rather impressive to see the number of variations that can be rung on the Viviani Theorem theme. See Polygon Altitude Problems II
I found this collection of related problems by James Tanton on Twitter. Even though all these problems do not involve perpendiculars, they have a common solution approach – a sort of theme and variations idea. In a later tweet Tanton refers to a Viviani Theorem associated with these types of problems. I did not recall that theorem explicitly or by name. I also have not looked it up yet, in order to solve these problems on my own. I am guessing there is a more classical Euclidean geometry proof, but I like my vector approach for its clarity. I also throw in a bit a calculus at the end for fun. See Polygon Altitude Problems I
This is a collection of simple but elegant puzzles, mostly from a British high school math teacher Catriona Shearer, for which I thought I would show solutions (solutions for a number of them had not been posted yet on Twitter at the time of writing). See the Geometric Puzzle Medley.
Apparently Catriona Shearer creates these problems herself, which shows an especially gifted talent. Ben Olin, of Math with Bad Drawings fame, had an interesting interview with Ms. Shearer. The reason for the interest in her work becomes evident the more of her geometry problems one sees. They are especially elegant and minimalist, and often have simple solutions, as exemplified by the “5 Problem” or “Shear Beauty” problem illustrated here. Words, such as “beauty” and “elegance”, are often bandied about concerning various mathematical subjects, but as with any discussion of esthetics, the efforts at explanation usually fall flat. Shearer’s problems are one of the best examples of these ideas I have ever seen. If you contemplate her problems and even solve them, you will understand the meaning of these descriptions.
One of the key aspects of mathematics is often its “hidden-ness” (some would say “opacity” or “incomprehension”). Her problems appear to have insufficient information to solve. But as you look at the usually regular figures, you see that there are inherent rigid constraints that soon yield specific information that leads to a solution. This discovery is akin to the sensation of discovering Newton’s mathematical laws underlying physical reality. It is the essence of one of the joys of mathematics.
A number of recent puzzles have involved perspective views of objects. I had never really explored the idea of a perspective map in detail. So some of the properties associated with it always seemed a bit vague to me. I decided I would derive the mathematical equations for the perspective or projective map and see how its properties fell out from the equations. With this information in hand I then addressed some questions I had about the article “Dürer: Disguise, Distance, Disagreements, and Diagonals!” by Annalisa Crannell, Marc Frantz, and Fumiko Futamura concerning a controversy over Albrecht Dürer’s woodcut St. Jerome in His Study (1514). And finally, I read somewhere that a parabola under a perspective map becomes an ellipse, so I was able to show that as well. See the Perspective Map.
(Update 7/1/2019) Continue reading
Sabine Hossenfelder wrote an excellent blog posting about the growing awareness that outstanding scientific problems are not getting solved at the same rate as in the past. Her whole article is worth a read, as are all her postings, but this latest contained a mathematical statement that warranted justification. For scientists “How much working time starting today corresponds to, say, 40 years working time starting 100 years ago. Have a guess! Answer: About 14 months.” See Hossenfelder Stagnation Problem.
This problem comes from the defunct Wall Street Journal Varsity Math Week collection.
“The coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?”
I changed the numbers a bit to make my calculations easier, but left the problem otherwise unchanged. When I checked the Varsity Math Week solution, I saw they used a simplifying formula that I could not remember. I also believed their solution left out a justification for the maximal area. Besides an intuitive solution for this, I also included a calculus version. See the Triangular Boundary Problem.
Yet another Futility Closet puzzle.
“Point E lies on segment AB, and point C lies on segment FG. The area of parallelogram ABCD is 20 square units. What’s the area of parallelogram EFGD?”
I had an alternative solution that I thought was a bit simpler and clearer. See the Parallelogram Puzzle.
This is another Futility Closet puzzle.
“Four straight roads cross a plain. No two are parallel, and no three meet in a point. On each road is a traveler who moves at some constant speed. If Blue and Red meet each other at their crossroad, and each of them meets Yellow and Green at their respective crossroads, will Yellow and Green necessarily meet at their own crossroad?”
I was not able to understand the solution given at first, so I tried to solve the problem on my own. Once I did, I was able to see what the Futility Closet solution was getting at. Certainly diagrams were needed to make sense of it all, and that is what I provided. See the Four Travelers Problem.
Reading Axios on Christmas Eve day 2017, I was struck by what appeared at first to be a strange graph showing preferences for Christmas movies divided between men and women. The thing that struck me as strange was the computation for the total votes: the percentages were the average of the men and women percentages. This, of course, is not how you average percentages. What was going on? See Strange Statistics.