The first objective is to comprehend the temporal congruence of all these thinking processes throughout history, which we claim that it is the decisive factor for the qualification of the diachronic character of mathematical theories. 

Instead of attacking directly the huge and impenetrable atemporal walls of the foundational schools attempting to glorify the diachronic character of mathematics, we adopt the strategy of bringing into light precisely these temporal unfolding processes behind the conception and development of major mathematical ideas in various fields, as well as their continuity and connectivity, independently of any rigid foundations. 

In this manner, we aim to embrace the obstacle erected by any type of atemporal foundations being unconsciously entrapped in the net of attempting to explain the diachronic by expelling time completely. Rather, it is the factual account of time that should be properly qualified and not confused with the full diversification of the temporal.   

Seeking for a conception of time going beyond the factual level, and being appropriate for providing a thread to investigate the development of mathematical thinking as a coherent living body, the major issue to tackle is the conceptualization of the present in this body. It is precisely the notion of the present that it is capable of providing insight into the diachronic character of a mathematical theory. 

The factual account of time is not suitable to capture the present in mathematical thinking because it depreciates it from its depth that extends bidirectionally both to the past through memory and to the future through anticipation and vision. This requires a novel syntaxis of time in the present of mathematical thinking bearing the capacity to bind appropriately together the past in terms of memory and the future in terms of vision. 

This binding process is not itself of a spatial static character, but it is intrinsically temporal and dynamic due to the bidirectional depth of the present. Spatiality may be attributed only to the traces of this process, whereas itself, pertaining to the depth of the present, and thus functioning synthetically and holistically, re-presents at each present the seed of resonance binding together harmonically memory and anticipation. 

In order that this becomes possible, the memory of the mathematical body cannot be static itself, in the sense that defining notions, like for instance the notion of a point, or the notion of a shortest line, require different associations in relation to the axiomatic and conceptual context of these notions. In a considerable amount of cases, the seed of binding resonance in the present involves the non-trivial abstraction of a notion defined within a certain axiomatic context to a novel one anticipating its role in a wider, or deeper, or more general and unifying axiomatic context. 

This novel syntaxis of time at each present of the living body of mathematical thinking endows representation with a temporal semantics, due to the bidirectional depth of the present, which does not bear the characteristic of repetition, but the characteristic of unfolding. Thus, semantic unfolding is the fundamental temporal criterion that we employ to qualify the development of mathematical thinking at each one of its presents under the non-factual syntaxis of time in the synthetic act of representation.

It is crucial to address the diachronic character of mathematical thinking under the proposed novel syntaxis of time at each one of its presents via semantic unfolding. Since representation concerns the depth of the present in its capacity to form a seed of harmonic resonance between memory and vision, the invariant characteristic of the present is subsumed in its bounding or encapsulating capacity with respect to different chords of semantic unfolding bearing the potential to resonate at this present. All these different chords extending bidirectionally constitute what we refer to as the depth of the present, whence their possible interweaving at the present realizing a seed of resonance makes semantic unfolding constellatory.

From this perspective, the role of an axiomatic framework cannot be of any foundational and constitutional significance.  Rather, an axiomatic framework serves as a scaffolding of possible resonances in the act of representation, in other words it serves as the resonator with respect to a motif at each present emerging out of constellatory unfolding. The encapsulating characteristic of the present is the invariant of constellatory unfolding that may assume different representations under change of scaffolding.