The Greatest Mathematician?

BaldFriede wrote: Let me just remark something: Today we don't have many problems thinking about the number zero; we are used to it. But to people in former times the zero was not an easy concept. Let me give you an example. There are three apples lying on the table. If all three are eaten, how many apples remain? You will say "zero of course", but there are also zero pears, zero cherries, zero peaches and zero pineapples, to name but a few of the millions of things of which there are zero. Why the heck should we especially name zero apples? Just because there had been three before? No, "zero" is not a concept we encounter in the world of things, it is a pure mathematical concept. Which is why the invention of this number was so very important for mathematics.

Yeah, right  

http://www.youtube.com/watch?v=3xewJOKyyoc

BaldFriede wrote: Let me just remark something: Today we don't have many problems thinking about the number zero; we are used to it. But to people in former times the zero was not an easy concept. Let me give you an example. There are three apples lying on the table. If all three are eaten, how many apples remain? You will say "zero of course", but there are also zero pears, zero cherries, zero peaches and zero pineapples, to name but a few of the millions of things of which there are zero. Why the heck should we especially name zero apples? Just because there had been three before? No, "zero" is not a concept we encounter in the world of things, it is a pure mathematical concept. Which is why the invention of this number was so very important for mathematics.

Very close to compelling argument.  However, there are some subtleties to discuss.  We should reply "zero apples" because we were asked about the number of apples (in technical jargon, the cardinality of the set of all apples on the table).  This is a well-defined concept (assuming that "apple" and "table" have sufficiently clear meaning, which I think they do for practical purposes).  Now you're completely correct to say that there are zero pears, cherries, etc., but those responses, while true, are irrelevant to the question posed. 

The cornerstone of scientific investigation is the isolation of variables of particular interest.  Of course, one might argue (with some legitimacy I think) that this is not a natural, but rather an acquired or learned, method of discovery or thinking.

Moreover, you are right to assert that "zero... is a pure mathematical concept" rather than something we "encounter in the world of things."  Yet this can be said of every number (have you ever seen a concrete "7" in reality, notwithstanding Sesame Street, of course?).  As physics models rather than describes reality, so does mathematics model.

I know a lot of this sounds like hairsplitting semantics, and practically it is, but these subtleties are crucial in precise and accurate reasoning about mathematics (or any other topic, especially abstract ones).