I'll start things off with this simple little example:

1 is defined as s0. (sn means "the successor of n")

2 is defined as s1.

Addition is defined as follows:

a + 0 = a for any a by definition.

a + b for b > 0 is defined recursively by a + sb = s(a + b).

With definitions out of the way:

Certainty in a formal system is amatter of definition, which is circular reasoning. 100% certainty becomes a matter of 100% agreementon definitions, which can never be universal or absolute.

1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2

Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?

To the extent that mathematics is abstract and completelydetached from objective reality, and if we are talking about simple arithmetic,then yes, because mathematical certainty is a matter of definition, but only when you assume 100% agreement about said definitions.

As it relates to reality, or even more complex mathematics, no,there is no 100% certainty. As itrelates to reality, certainty is a matter of inductive reasoning, which isnever 100% certain. As it relates to mathematics that is more complex thansimple arithmetic, certainty is a matter of deductive reasoning and thevalidity of the associated axioms.

Euclid’s 5th postulate was alwaysquestionable because it cannot be derived from the other four, nevertheless, it was accepted that deductive reasoning led to certainty for centuries. With thediscovery, or invention, depending on your point of view, of non-EuclideanGeometries three of four times in the mid-19th century, the two-thousand-yearbelief that deductive reasoning led to Truth was lost. With multiple deductively perfect geometries,the only way to determine which is true of space, becomes a matter ofexperiment and measurement, which gets fuzzy and eliminates any chance of 100% certainty.