Mathematics is rife with contradictions, is shot through with them: the fault-lines lie where theories collide, fade or open up.
Does this disturb the incarnation of mathematics - the Ideal Mathematician?
The Ideal Mathematician is not. His inward calm being the Being of Mathematics, his third eye cast on the third realm - the third realm of pure abstractions. Unique in themselves and as Singular as crystals and more Splendid; the shadows of which treads his own human mind - the secondary realm of cognition and the intellect.
(Can one say, that mathematics is found in the primary world, the realm of actual physical being? Possibly it is obtained as a correlation between them).
From the Purity of the third realm, comes the light, that places all things in their correct station and gravity, for in the Third Realm, it is already as such. For in the Third Realm contradiction does not obtain.
Thus that Spiritus Mundi of Mathematics - Platonism as Plato did not put it but as co-opted - is not troubled by vast images of discord and diction, they are illusions of our too-human mind.
Say, as a hypothetical, that Platonism is also an illusion, that the Being of mathematics is not-there. Then what is left? Not its Being, but its Becoming, and its Becoming must involve contradictions essentially. For there is no appeal to the Ideal to whisk them away. And these contradictions are not on the frontier of mathematics (were we to imagine mathematics as some vast ascending Sphere) but reach back right in and through and back and in through the body of mathematics (though we have banished the Body) - in every place and in every direction. Thus mathematics as Becoming unfolds.
Here is the Event of Mathematics - its adventure.
Does denying the Being of Mathematics (Platonism) leads necessarily to pure Becoming, and in the Being of its Becoming (and not the Becoming of its Being) are contradictions essential - that is irreducibly, inerradicably irremovable?
Given some of the comments about the obscurity and opacity of the language, I thought it might be useful to 'explain' the question.
The mainstream ontology, I take it, of mathematics is Platonism where abstractions like the number '2', or the group 'Z x Z' exist; but moreover that propositions about these objects also exist with well defined truth-values. That is the proposition '2 is an even number'; and further that theories themselves also exist, like PA with first-order classical logic. The Platonic realm, is considered to exist outside of space & time. I also consider it mainstream that the law of the excluded middle or that contradictions are possible in this realm are not possible. (Notably, in Aristotles discussion of the law of contradiction, he left open the possiblity of what kind of truth value one can assign to a proposition that refers to the future. But in the Platonic realm there is no time, so no future).
I ask, suppose that Platonism isn't true, and that at least time is inherently involved in the ontology of mathematical objects, if not also space. One might position it as taking the epistemology of mathematics as its ontology.
I then discuss the role of the law of contradictions when this is done. One might say that what contradiction means in epistemological terms is different from what it means ontologically. And I'm proposing that contradictions are essential epistemologically; because unlike Platonism, where truths are exhibited all at once; epistemologicaly, there are different theories, some of which whose propositions or theorems they may have in common, others may contradict. It may be true, that in the process of time, one may bring these theories in line with each other, but I also expect that very movement also brings into view other theories which are incommensurable.
I pick-up Whiteheads terminology for the Platonic realm - the third realm and take the first & second to be Descartes picture of the world as being divided into physical & mental substances. I refer to the ontology of Mathematics, as its Being, alluding to Badious conception of Mathematics as the 'very site of ontology', which is a resucitation of Platos ontology of Forms, but with the Forms as considered as abstract objects - this is (very) different from mathematical Platonism.
That I talk about Purity and Light is an allusion to the emanationist philosophy of Plotinus which is indebted to Platos Philosophy; but also interpreted in the question as the 'lack of contradictions', and also as a metaphor for the worlds to influence each other; but also so I can bring in Platos famous picture of his Forms casting 'shadows' in the real world.
Becoming is to Being in Continental thought as the Heracleitian Flux is to Platos Forms. Hegel refers to Becoming as the sublation of Being & non-Being, and Heidegger when thinking of Being qua Being, identifies Being with time, that is Becoming.
Though I mention the 'Event' of Mathematics, its 'Adventure' as where new ideas 'collide, fade or open up' and refers to Mathematics as an active and creative process, its not the main intent of the question.