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The aims of the project, as seen by a philosopher December 15, 2007

Posted by Alexandre Borovik in Uncategorized.
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Any intellectual activity is governed by freedoms and constraints. Mathematics is often taken as a discipline which, while requiring obedience to strict standards of rigour, allows the practitioner the freedom to develop ideas in whichever direction he or she pleases. For Georg Cantor,

“The essence of mathematics is freedom”

and much of the philosophical worlds believes him, seeing only some or other logical system as offering a boundary to a mathematician’s work.

But others have recognized that to be perceived to be a good mathematician requires much more than this. Not just any logical truths can be produced. One’s work must be one or more of the following: surprising, illuminating, revealing of new vistas, coherent with what has gone before, able to link apparently distant theories, applicable. Appreciation of this aspect of mathematics fits well with Hermann Weyl’s assertion that

“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”

Now it might be that while the restriction of logic is timeless, other restrictions are more like fashions. Perhaps the mathematical culture determines that what won’t  be recognized today, will find an audience tomorrow.

Or perhaps it is our psychological faculties which largely determine opportunities for us, while also standing in our way. Think of how much effort was required to liberate our geometry from the grip of Euclid, and why we study knots in three-space before turning to knotted spheres in four-space. More positively, perhaps it is our psychology that provokes us to construct concepts which will allow us to control a field.

On the other hand, it can happen that what seemed to be temporary interruptions to our progress in a field are actually due to what is structurally possible. Mathematical results themselves later reveal that what frustrated mathematicians’  attempts to achieve a certain goal lay in the nature of mathematical structure itself.

In this project, we aim to bring to bear these considerations onto the question of infinity. Where Cantor saw untold riches in the world his set theory opened up, especially as concerns infinitely large objects, what we find instead is that mathematicians more than a century later employ a limited range of such entities. As Weyl remarked, set theory contains too much “sand”. But are mathematicians just being culturally conservative, are our minds not geared to go beyond these familiar types, or are there metamathematical reasons for such constraint?

Perhaps our psychological needs dictate that the only uses of infinitely large objects we employ are those which allow a certain cognitive control on a field. And perhaps this cognitive control is best explained by our need to tell a story about that field. Then if only a very few entities allow such a story to be told of them, we should expect this limitation. But then, is this capacity to tell a story not just a sign of current cultural limitations? Will our descendants look back at our limitations, as we do our ancestors who could conceive of no other geometry than Euclidean? Perhaps a new language will liberate us once more, opening up possibilities we have yet to consider and making formerly implausible ones seem to be within our grasp.

We intend to explore these issues by historical analysis of episodes in the emergence  of mathematical and philosophical conceptions of infinity from the Greeks to Cantor and beyond. We shall examine metamathematical theory which bears on the question of usable infinite objects. We shall consider the use of infinite objects in actually practiced mathematics and computer science.

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