Here is the question:
How is it that POE [the principle of explosion] does not make mathematical logic inconsistent?
If the axioms of mathematical logic were inconsistent then the principle of explosion would reduce mathematical logic to trivilism where all propositions are true since they can all be derived from that inconsistency.
However, having the principle of explosion does not make the axioms inconsistent. That would require a separate step. To show that the axioms are inconsistent one has to derive some proposition P and its negation ~P from those axioms (not from some other assumptions) prior to using that principle to derive all propositions.
Here is how Wikipedia represents that symbolically:
P, ¬P ⊢ Q
The important thing to note is that we have to first derive both P and ~P. Then we can use explosion to derive Q.
Reductio ad absurdum is a way to derive the negation of an assumption should that assumption lead to a contradiction. Here is Wikipedia's description of it:
In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments or the appeal to extremes, is a form of argument that attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible.
That assumption is not one of the axioms. It is only tentatively assumed. That one can derive a contradiction from that assumption and other propositions is often what one desires to do so one can derive the negation of that assumption.
The OP adds this question:
Why is the Principle of Explosion (that starts with a contradiction) allowed when it is common knowledge that inference stops when reaching a contradiction?
One use of explosion in a natural deduction system is to handle one or both cases of a disjunction to derive a result using disjunction elimination (vE). Here is a proof showing how that works.
To eliminate the disjunction in line 1 and derive the goal P, I have to consider two cases: P and Q. The first case, P, is easy. It is already the goal I want to derive. The second case, Q, is more difficult. However, I also have an assumption on line 2, ¬Q, that I can use to derive a contradiction, symbolized by ⊥ on line 6. Now I use explosion on line 7 to derive P from that case as well. From explosion I could have derived anything, but what I need is P. So I derive that.
Having derived P in both cases I can use disjunction elimination to eliminate the disjunction on line 1, referencing both cases on lines 3-4 and lines 5-7 and derive P on line 8.
This would be one use of explosion in a natural deduction system.
Wikipedia contributors. (2019, August 15). Reductio ad absurdum. In Wikipedia, The Free Encyclopedia. Retrieved 13:25, September 11, 2019, from https://en.wikipedia.org/w/index.php?title=Reductio_ad_absurdum&oldid=910921053
Wikipedia contributors. (2019, September 6). Principle of explosion. In Wikipedia, The Free Encyclopedia. Retrieved 13:21, September 11, 2019, from https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=914233106