I give my understanding of the place of Topos theory in mathematics :

Traditional mathematics are based on both the langage of set theory and classical logic . This means that any mathematical object is then described as a set with elements which becomes more and more complex as its level of abstraction increases.

Topos theory is issued from the langage of categories. More precisely Topos realise a kind of minimal interpretation of set theory in the langage of categories. This new view on sets gives a lot of freedom on the properties of the universe of sets on which the mathematician intends to works. For exemple it is claimed possible to consider sets without having to deal with the notion of elements of this set.

It seems that it is easier to cope with mathematical abstraction in category theory than in classical set theory. This is, may be, one part of the explanation for the vast amount of conceptual creation in mathematics brought by the German/Ukrainian/French mathematician Alexander Grothendieck.

It would be interesting to know from C. Isham if these considerations are important or not for the use of Topos theory he envisages in Physics

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