Although not all accepted his proposal, uncompromising republicans like Brugha and Stack, though not enamoured with it, were eventually won over by de Valera's mathematical explanations, which were based on set theory.

During the Dáil Éireann debates, de Valera drew a Venn diagram to explain his theory and explained it as follows:

Although each of the states is represented by a circle, which would lead one to believe that they were sets, they were more likely to represent elements. Therefore, though it may appear that each country is a subset of a larger set called the British Commonwealth (BC), the countries are probably elements of the set, BC.The British Commonwealth was represented by a large circle within which were five smaller circles, each representing one of the self-governing countries of that group of nations. The President[de Valera]sketched in Ireland as a circle outside the large circle but touching it.

Some difficulties arise when one tries to decide if de Valera meant the sets to be open or closed. A circular disc which is open does not contain the circumference; however, a circular disc which is closed contains the circumference, i.e. (BC, Ire) does not contain the tangent point; however, [BC, Ire] does. It is unclear whether he viewed the discs as open or closed.

Thus, it is difficult to ascertain whether the intersection of {Self-Governing Commonwealth Countries} and {Ireland} is empty or non-empty. If we are to assume it is a closed set and the point where Ireland is tangent is common to both, is this what de Valera envisaged when he proposed External Association? If so, to whom does the point belong, if it is a physical entity? Does it belong to both equally or not? In political terms, did de Valera want the connection to exist, but have no meaning or power? The idea he was trying to explain was that Ireland would be *associated with* the Commonwealth but not a* member* of it, i.e. Ireland would be tangent to the Commonwealth, but would not have any elements in common with it and most importantly, would not be an element of the Commonwealth as was the case with the other five states mentioned.

This is the most common example which was reported in the media and covered in biographies, but de Valera also drew other diagrams, depending on the political perspective of his audience. At the time, set theory was still a relatively new branch of mathematics, and though it may not have been covered in schools, it seems the Dáil members understood de Valera's explanation well enough to vote for it. Though de Valera had not been studying mathematics formally for some time, through instances such as this one it is clear he still used mathematics as a tool for solving problems.

Article by: Cáit Ní Shúilleabháin, University of Cork, caitnishuilleabhain@gmail.com