This is a question that a 5-year-old would be laughed out of the Kindergarten if he could not answer, but on which it and related questions Bertrand Russell spent 10 years of his life. You don’t need to look further than pure mathematics. Mathematics is a bit like an onion. Even research mathematicians may never reach the core of all areas they use. Betrand Russell was one of the few that studied the fundamentals. So at school one may peel back a couple of layers of the onion, at University a couple more.
Another problem which usually goes undiscussed is definitions. Most mathematicians you talk to claim that mathematics has only precise definitions. Well, the truth is that the definition is in the eye of the beholder. Mathematicians would like them to be, and it is true definitions get more and more obscure as you peel back the onion, but for the most part definitions are not totally abstract entities with a universal non repudiable meaning. Russell’s work is probably the nearest to this but except for a few extracts who has read his Principia Mathematica? Most definitions in mathematics are targeted for a specific audience, more precision is added as the sophistication of the audience grows.
When a teacher asks her/his 5-years old class “what is one and one” or what is one plus one”, usually “and” and “plus” are interchangeable, they have one picture in their minds, so the question becomes unambiguous. Very soon they will see this more graphically as: what does 1 + 1 =. This will become known to them as “doing sums”. Quite often they are given a list to fill in: 1+1= ?: 1+2=? etc. Unfortunately this is very sloppy thinking which might be appropriate for 5-year-olds but is never corrected later in the schooling.
So what is wrong with asking what is one plus one or asking what number does 1+1=? Well we can assume we are using the decimal system, admittedly a not properly defined definition. So 2 is a perfectly good answer but so is 1+1. That is the point, 1+1 is a perfectly good number. This may be more easily seem if you asked what does e + pi =? Well the only perfect answer I can give is e + pi = e + pi. neither can be written down nor combined. In fact what does “=” actually mean?
At some stage during our schooling we should be introduced to the successor theorem so that we can understand that all the natural numbers are created from the identity (1) plus applications of the successor algorithm. We then won’t confuse the use of “0” as a placeholder with the number “0” itself.
Step-by-step explanation:
