April 21st, 2007 infitessimal algebra

In physics and intro calculus courses, the symbol dx is taught to be a literal infitesimal value. Hence, the derivative, dy/dx is considered literally to be a ratio of infitessimal values. The integral, with a dx at the end, is thought to be highly analagous to the summation, with the infitesimal term simply serving as another factor in the “infinite summation”. This sort of thing can be taken to the extreme: to calculate the derivative of a function, simply plug in x+dx into f(x) and do some algebra, keeping in mind that dx^2=0 but dx itself is not zero! Surprisingly, that gives you the right derivative! Of course all this is not rigorous at all. Doing algebra with infitessimals is problematic on many levels. There are many problems both mathematically and philosophically, not the least of which is the law of excluded middle.

But the fact that infitessimal algebra “works” suggests that it should be possible to make it rigorous. And indeed, this has been done by many mathemticians. One example is called Non-Standard Analysis, where the Real Numbers are extended to the Hyper-Real Numbers, where infitessimal values are added to the Real Numbers. Apparently, you need to deny the law of excluded to make this work though. Many mathematicians are also somewhat skeptical of the topic. Some have critisized that the theory suffers from ontological problems.

What is more interesting is that when Newton and/or Leibniz first invented calculus, they did so mostly using an algebra of infitessimals. Hence, their invention was not so rigorous or well-defined. It took work by many other mathematicians years later to do the “cleaning work”. Now, it is possible to make sense of infitessimal algebra without a rigorous theory if you make some ad-hoc arguments, and I suppose this is why it continues to be used in classes. I don’t think I have a problem with this, as long as some of the flaws are pointed out as well. If some mindless high school teacher is teaching calculus without all the philosophical implications of infitessimal algebra, it’s a shame.