the way we do it or the way it should be done

Question

in the preface to one of his works Sir Bertrand Russell writes:

.. in mathematics the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point ..

isn't it the contrary .. is it not at the very start, in the choice of the primitive ideas and primitive relations making up the axioms, that we usually find the greatest degree of self-evidence?

He then goes on saying:

.. hence the early deductions, until they reach this point, give reasons rather for believing the premisses because true consequences follow from them, than for believing the consequences because they follow from the premisses ..

does He mean then that such greatest degree is not found BY US at the beginning , thereby supposing it should be?

Answer 1

In a nutshell, the thought of Russell is : in mathematics the axioms are not always self-evident.

Consider e.g. Peano axioms; with them we prove all arithmetical "facts" included the very obviuos ones : $1+1=2$ and $n+m=m+n$.

The benefit is: to show that the huge number of arithmetical laws are all theorems that cab be proved from very few basic principles assumed as axioms.

See Bertrand Russell, Introduction to Mathematical Philosophy (1919), page 5:

He [Peano] showed that the entire theory of the natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic.

The development of the axiomatic theory of numbers was a late development: it started around the middle of 19th Century with H.Grassmann and was completed by the end of the century with R.Dedekind and G.Peano.

This happened after a large body of concepts and theorems about natural numbers has been accumulated during the centuries.