Posted by: Alexandre Borovik | January 31, 2010

## First Proper Mathematics

On a large sheet of paper I made a triangle of numbers and addition signs as below.   Down the right side I made a list of the results of the additions.   It was clear that this process could continue as long as I wanted and my attention went to the vertical sequence on the right.

.                                                                                            0

1                                                                                           1

2 + 2                                                                                  4

3 + 3 + 3                                                                          9

4 + 4 + 4 + 4                                                                16

5 + 5 + 5 + 5 + 5                                                        25

6 + 6 + 6 + 6 + 6 + 6                                                 36

7 + 7 + 7 + 7 + 7 + 7 + 7                                       49

…….                                                         ..

It was clear that the numbers in the sequence increased more rapidly as you went down so I formed the sequence of first differences.   Of course I obtained the odd numbers.   So I thought, “Is this true in general?  Does the sequence continue always to generate the odd numbers no matter how far we go?”   I also thought, “Can I prove it?” and asked my father, who had a PhD in chemistry.   He confirmed that the odd numbers were indeed correct and mentioned algebra.   I wondered how can he know and can I prove it?   I think I thought in terms of a proof based on counters; I did not know my addition tables and certainly not my multiplication tables, and performed the additions by counting, mainly in my head but possibly also using my fingers.   I did not properly formulate a proof based on counters until grown up, as I later had algebra that made the result obvious anyway.   A proof based on counters is quite easy and possibly I got near to it at the time.

Perhaps I did not continue thinking about the matter to the point of constructing a proof because I became aware of the question, “Even if I get a proof, how will I know the proof is correct?”   This question bothered me.   I think I was aged four at the time, coming up to five, just after the Second World War was ended.

The point at the top of the triangle denoted zero zeroes added together.   The symbol “0” would not have been correct and I had a little difficulty deciding what I should put at the top.

Evidently I understood zero.   At some point, probably earlier than the research, I had discovered that you can continue counting forever, using the usual representation of numbers if one ran out of names.

My mother tongue is English and the above mathematics was all in English.