# Category Archives: Philosophy of Mathematics

## Between Feynman in Babylon and Metaphysics: What the Mathematical Process and the History of Science Can Tell Us Philosophically about the Education Process

Since I have spent the summer studying mathematics at Harvard University with Jameel Al-Aidroos (Ph.D Berkeley), expect that my next few posts, or at least some of them, will be on topics related to mathematics.  I want to take some time, in this blog post, to look at where mathematical thought fits into some of my understandings of I have gleaned from studying the history of science.  The upshot of the historical, philosophical, and mathematical content and musings will be pedagogical, just to give the reader some idea of where I am going.  An important thing to understand, before reading this post, is the distinction between pure and applied mathematics.  “Pure mathematics,” as opposed to “applied mathematics,” is, in its essence, math for its own sake, entirely apart from possible applications.  In many cases, pure mathematics initially has no known application.  Additionally, pure mathematics deals with abstract entities that have been detached from particular entities —and this will prove to be important to what I will say later.

## Why .9999… (Repetend) Is Not Equal to 1

I recently bumped into a graduate student in the economics department at the University of Pittsburgh, Shawn McCoy, and he brought to my attention that there are some folks who wish to claim that .9999…=1.  That is, the decimal value, .9-repetend, which has infinitely many places of ‘9’ after the decimal, is equivalent to the whole number, 1.  Any individual of sufficient commonsense and no real inclination toward contrarianism-for-the-sake-of-contrarianism will maintain that the claim is silly and move on.  However, there is a bit of mathematical prestidigitation —and that’s precisely what it is, as I will show— presents an “argument” to the contrary of commonsense.  The argument requires that we do the following:

Let x be .9999…  Then, let the left-hand side (LHS) of our equation be 10x-x.  Also, let the right-hand side of the equation (RHS) be the same, not in algebraic terms, but in numerical terms: 9.9999…-.9999…=9.  Solving the LHS, we get 10x-x=9x.  The conundrum is that it should be the case, of course, that LHS=RHS.  However, if one divides LHS and RHS by 9, the consequent value is x=1, though, ab initio, we said that x=.9999…  Therefore, some try to conclude, .9999…=1.  Continue reading