Traditionally the notion of ‘reality’ is supposed to capture the mind-independence of what is to be known. If science is a form of knowledge, it is thus supposed to deal with a corresponding form of ‘reality’.
Now, mind-independence is a very generic concept and we should certainly individuate it according to the activities of the mind in relation to which it is considered. The shape of reality is not the same considering the kind of ‘knowledge’ at stake.
According to an entrenched picture of mathematical objectivity, it consists of entities that exist by themselves, independently of the mathematical operations and procedures, and that the mathematician, as it were, ‘discovers’.
We can generalize this picture and make it the paradigm of a realism that we should call ‘Platonist’ – in the sense in which this word is commonly used in the philosophy of mathematics.
In contrast to that position, constructivism questions the very notion of reality, holding that the object is the mere product of our mental activity. On that side as well, mathematical constructivism can be made the paradigm of a general ‘anti-realist’ ontological attitude that goes far beyond the realm of mathematics.
Our purpose will be to investigate whether any meaning is left for the concept of reality outside any Platonist framework. Does realism consist, in mathematics as in general, in itemizing   entities as they are supposed to be given by themselves–beyond any procedures? Or where do we face ‘reality’ and become able to make sense of it but in our very operations and ways to proceed?
The mathematics of the twentieth century constantly investigated this question. Not only the so-called quantum mathematics where this debate  is structurally central, but the fascinating development of probabilities or the evolution of the concepts of solutions of partial  differential equations, to give very few examples, reveals the importance of this intrinsic refoundation of the concept of realism in  mathematics. And therefore the traditional position of Platonism seems to have been relegated to a secondly role by the active mathematics (but not by all active mathematicians, unfortunately).
Thus, in an open discussion between philosophers and mathematicians, we shall explore the paths of a possible ‘realism without   entities’ – that is to say that does not primarily consist in positing   entities, that could not bear any absolute meaning, but in analysing how and with which import our ways to do define possible formats for such ‘entities.’

The workshop is partly funded by CNRS Paris

 http://www.math.polytechnique.fr/~paul/phenomath/

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