The laws of mathematics

There should be no question of using one’s judgement or intuition in applying the rules one learns in math class, of course. . .

We’ve noted that some sets of rules are best applied not very strictly, and that even scientific laws often need a bit of judgement in their use.  When we come to mathematics we expect to have no such leeway.  Mathematics is the rigorous, consistent structure it is by having carefully proved everything (except a few necessary postulates); judgement should not come into the process.

Take, for example, multiplying two numbers a and b.  If a x b = 0, then either a or b (or both) must be zero.  This is the basis for solving polynomial equations by factoring.  Though the rigorous proof is not normally given in High School texts, no one seriously doubts it.

But even in High School students learn about things called vectors.  You can multiply them two different ways, with the dot or cross product.  Without going into detail, we’ll just point out that two vectors, neither of which is zero, can have a vanishing dot product (if they’re perpendicular) or cross product (if they’re parallel).  You can’t apply the earlier rule to vectors.

Students also learn, early on, that you can multiply numbers in any order; for any a and b, a x b = b x a.  Our tutoring consultant sometimes has to point this out to students struggling with simplifying some recalcitrant expression; we’ve done it so long we no longer think about it.

But again, some High School students (and more in college) learn about matrices, certain arrangements of numbers or variables; for them, it’s a special result if AB =BA.  The matrices are then said to commute.  Whether a given pair of matrices commutes is a very basic question in Quantum Mechanics, with extensive consequences.

Let’s go on to Geometry.  The angles of any triangle add up to 180 degrees; or if you want to be purer and leave out numbers, two right angles.  This is the basis for many a practical problem in dividing up a surface, and if you fix one angle at 90 degrees, for all of trigonometry.  But it was already realized in antiquity (though maybe not explicitly) that on the surface of a sphere triangles will add up to more than 180.  Negatively curved surfaces, where triangles add up to less than 180, were discovered in the nineteenth century.

The Pythagorean Theorem?  It, too, only holds on a flat surface.

There is one more bit of relearning we’ll point out.  Suppose one has an equation that looks like x = x + 1.  This makes sense in computer languages, but we’ll stay in mathematics for now.  The standard way to deal with it is to subtract x from both sides, obtaining 0 = 1.  This is not true; therefore, there is no x that will solve this equation.  Vectors and matrices will not help us.

But if x is a transfinite number, like the number of all integers, or the number of points on a line (these are different transfinite numbers), the equation is true.  The number of all integers starting with 2 is the same as the number of all integers starting with 1.

So you do, after all, need to apply the laws of mathematics carefully.  That’s why mathematical statements generally start with a list of specifications of the things they’ll be working with.

And we suspect that a lot of mathematical research begins with the question, “What if this isn’t true?”