“dialectic is not part of mathematics,
and its notions are unrelated to the primitive notions of a theory.
We have already deﬁned, in our thesis (Lautman 1938b),
the priority of the dialectic
as that of ‘concern’ or the ‘question’
with respect to the response.
It is a matter of an ‘ontological ’anteriority,
to use the words of Heidegger,
exactly comparable to that of the ‘intention’ in regards to that ‘for the sake of’.
Just as the notion of ‘for the sake of’
necessarily refers to an intention oriented towards this purpose,
it is of the nature of the response to be an answer to a question already posed,
and this, even if the idea of the question comes to mind only after having seen the response.
The existence of mathematical relations therefore
necessarily refers back to the positive Idea of the search of similar relations in general.
Having established this point
allows us to specify that which constitutes the essence of the Ideas of the Dialectic.
First of all let us point out that we distinguish
The Ideas envisage
possible relations between
Thus we examined in our thesis (Latman 1938b)
the Ideas of the possible relations
between pairs of notions
whole and part,
situational properties and intrinsic properties,
basic domains and the entities deﬁned on these domains,
formal systems and their realization, etc.
The essential difference between
the nature of mathematics
and the nature of the Dialectic
can be inferred from these deﬁnitions.
While the mathematical relations describe
the connections that in fact exist between distinct mathematical entities,
the Ideas of dialectical relations are not assertive of any connection whatsoever
that in fact exists between notions.
Insofar as ‘posed questions’,
they only constitute a problematic relative to the possible situations of entities.
It then happens to be once again exactly as in Heidegger’s analysis,
that the Ideas that constitute his problematic
are characterized by an essential insufﬁciency,
and it is yet once again in this effort to complete the understanding of the Idea,
that more concrete notions are seen to appear relative to the entity, that is, true mathematical theories“