It’s True. There’s a Proof. Here’s Why.

I was reading through Topology Now! by Robert Messer and Philip Straffin, a really good book with a unique expository structure which I have read twice, when I came across a proof that there exists “a real number whose square is .”  The demonstration was in the text to show an application of the Completeness Property of Real Numbers.  At one point in the proof   is chosen such that . As is common in proofs, the expression seems to come out of the blue.  By this I mean that, as becomes clear  three lines later, the expression for is exactly the value that makes an inequality true and leads to  a contradiction. But there is no logical process I can see by which the bound on could have been discovered before making a trial of those three  steps.  The  must have been found in the process of developing/ discovering the proof and did not form out of thin air as it seems in the formal proof.

So there does exist a real number whose square is .  We have a proof – a formal argument consisting of eliminating cases using  proof by contradiction.  We can check the proof for hidden assumptions, algebra errors, missing cases etc.  If everything is okay, we have a new fact for our later use.  But what have we learned? It depends on the proof.  There may be obvious generalizations or a new technique to put in our tool box or as in the example at hand some insight into a mathematical structure.  But do we really understand why is a real number?

One can imagine “teaching” the proof – emphasizing the crucial assumptions, outlining the structure, anticipating small difficulties.  This would be an exercise in modeling how to read a proof. Or we can ask “Why is the statement of the theorem true?”  If we had a model in our heads that helps us understand why, maybe  we can extend the statement to say or .  The model will be  individual. Each of us has different neuronal models for mathematical concepts that we use for our own understanding and which we must have before we can explain ourselves to anyone else much less prove something.  For the idea that the square root of exists, we might look at the length of the hypotenuse of an isosceles  right triangle or consider a series of rational approximations to or think about the “empty” space on the number line.  Building these models, that is, conceptual understanding of mathematical structures is much harder than understanding the steps of a proof.  This is why we have math instructors and math courses.

Getting back to Topology Now! The particular proof I happened to read was intended to illustrate the power and necessity for The Completeness Property.  It serves its purpose.  This post is not a critique.  I just happened to read it and my mind took off in this flight of fancy.

The Urge to Sharpen a Pencil

A few moments ago I happened to notice a small pencil sharpener, the kind with a steel blade set at angle, in my tray of miscellany.  I was struck by an urge to sharpen a wooden pencil.  I wanted to feel the blade cutting through the incense cedar fibers.  I felt this need physically, starting in  my front shoulders and forearms,  to make some wooden shape with a smooth cut of a perfectly sharpened chisel or knife or plane or handsaw – a throwback to my carpentry days.

In the same way, as I watched our senior math majors’ capstone talks, I felt the need to understand some small area of mathematics and explain it to somebody as they were doing.  This is, of course, part of my job, what I do, a fun part.  The emotional imperative to explain and explain well might account for whatever quality my lectures have and  comes from the depths of my being.

Vocational Education

Joe Klein has an article in Time magazine (May 14, 2012) entitled “Learning that Works”  on vocational education.  The most arresting information: Students in these programs do better on state comprehensive tests, graduate from high school at a higher rate and go on to post-secondary education also at a higher rate than nonvocational students.  Of course.  If for no other reason than CTE (Career and Technical Education) courses immerse their students in the physical world – an environment that provides immediate feedback and accepts no excuses.  Mis-suture a sheep or leave a bolt untightened and the consequences are harsh.  The world of a teenager is physical – tactile and kinetic.  Young people are in optimum condition for learning  any hands-on practical knowledge.  And the concrete world enforces its own discipline requiring close attention, consistency, planning and self-confidence.  These traits are also necessary for success in life and for success as a student of any kind.  So, of course Voc Ed students can be and are also successful in their academic endeavors.

On a personal note, a zillion years ago I dropped out of graduate school.  As I now understand it, I was depressed.  I call those few years of knocking around the country my “hippy years” without the drugs.  But really that was the time I found myself – at least myself in the physical world.  I worked in factories, on farms and in restaurants and any other place that would take me.  I learned to cut felt, dry tobacco, and cook Chinese style.  I learned to knit, to sew and to repair my possessions.  I particularly acquired good carpentry skills and developed enough confidence to build a stone house and also enough confidence to stand up in front of a classroom of strangers and start my teaching career.

Dickens and Mathematics Somehow

I’ve noticed that lately my blog posts have been becoming more personal.  Maybe it is a phase I’m going through.  Anyway I just finished reading Dickens’ Little Dorrit. Besides the well-known social motifs relating to poverty, prisons, bureaucracy, and bubble economies, I detected the theme of shifting reality. As one example the particularly arresting chapter “On the Road”  has little Dorrit drifting along the canals in Venice and seeing or at least sensing the Marshalsea (debtor’s prison) in every tableau.  Is it possible that one can accumulate enough meaningful memories that one has a certain sense of unreality or at least a disconnection to present experience?  Can our accumulated past life over-filter the present?  For those of us who spend a lot of time thinking (There I made a tenuous connection to doing/teaching mathematics.) is this a natural state?  We spend so much of our  time problem-solving, observing our own actions for correctness and clarity, checking our assumptions and searching our memories that we might lose touch with the immediate present.  Is this my introverted nature justifying itself?  Or my aging self?  Or are shifting realities just part of the human condition?

Nobody Calls Me Anymore

I am getting fewer and fewer phone calls at my office nowadays.   My telephone number is published on the syllabus and on the class website and I continually urge students to “give me a ring” if they have a question or a problem, but the phone never rings and the red light never blinks.  In the current Atlantic magazine (May 2012) Stephen Marche states in “Is Facebook Making Us Lonely?” that “our omnipresent new technologies” – I would include texting as well as facebook – help us avoid the “mess of human interaction.”  That is precisely it.  My students seem to be more comfortable with the extra distance that texting and facebooking imposes on them.  They seem less comfortable with direct contact.  I get few office visits or even emails from them.  Right now I neither text or facebook.  I think I have got figure texting out.  Is there a texting system that goes to email?  Was that a dumb or naive or clueless question?

Math Is More Than Just Algebra

I alluded in a couple of earlier posts, here and here, to the fact that our high school mathematics curriculum is in a rut focusing mostly on basic algebra and that many algebra skills will only be useful in calculus and the physical sciences.  I also mentioned that there are plenty of cool topics in mathematics that are not algebra based.

Recently I ran across the project below that explores the concept of minimal spanning trees.  I constructed it as part of an assignment for a weekend class at Chico State University that I took many years ago.  At minimum it can be treated  as just an exercise in addition.  I like it because it exposes students to a non-computational algorithm and easy but clever proof.

Minimal Spanning Tree Exercise

Did I Have a Mean Mom?

We always made our beds.  We always said, “May I be excused?” before leaving the kitchen table.  We produced spotlessly clean windows and trimmed all the edges when we mowed the lawn.  Our mother set a standard and we attempted to perform to that standard “or else.”  Sometime this required, how shall I say it?, strong language from my mother.  She was not tiger mom, my mother Josephine Hatton, at least not in the sense that she was preparing us for success in an adult life.  She just enforced high standards of performance by her children and  I now understand how hard that was on her.

We never saw mom’s gentler side though she had one, particularly in relation to my father, but I know her perpetual attention to detail must have taken a toll.  I know this because I feel the same  tension (as far as I know) when I attempt to hold my class to high standards – in their writing, in the completeness of their knowledge, and in enforcing a certain level of fairness.  I feel sometimes that I am swimming upstream against currents of unaccountability and looseness.  I makes me seem mean and I don’t think I am mean.  My mother wasn’t either.