Transcendental Knowledge?

[Mudhammam]

Just because we only know the world by means of the senses does not mean that you cannot reason about the parts of reality that exist apart from the senses. Form this process someone can know of real things that exist apart from particular knowing subjects. For example, the truthes of mathematics are discovered, not invented. They are necessary truths that do not depend on a knowing subject for their validity. Secondly, perceptions are contingent upon the existence of an independent perceiver. The perceiver must exist prior to its perceptions, unless you posit a perfectly self-perceptive preciever ;-) or a real unifying and insensate principle that binds experiences into identifiable knowing subjects.

Necessarily true as in a transcendental truth?
I take that to mean that true mathematical statements are necessarily true insofar as they relate to certain dimensions of space (three, for sure, I don't know a damn about 11-dimensions) and change that is successive (as in time)... because anything else is simply inconceivable to me. The fact that mathematics break down in the case of Singularities, however, casts doubt in my mind that even mathematics, though accurately descriptive about the reality within which our brains evolved to take advantage of, itself can be said to contain explanatory power, or necessary truths, about more fundamental levels of reality.

Is it true that the object of consciousness ceases to exist apart from a knowing subject? Again I would refer to the truths of mathematics whose objects are immaterial and thus can be known by the intellect apart from any particular sensible form.

I would agree. I'm not sure if idealists would suggest that objects of consciousness wholly cease to exist apart from a knowing subject, only that they're existence as subjectively perceived would cease to be...leaving something analogous to objects, that is, completely unknown, in a state of infinite potentiality? Again, I'm a bit unclear as to what Idealists think exists outside of perception (there seems to be different strains or degrees of idealism).

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Ignoring the wall of text and focusing on the thread title, there's no such thing as "transcendental knowledge" (or "transcendental" anything, really, but that will become obvious in a second).
"Transcendental" means that it transcends everything. Your thread title literally means "Knowledge that transcends knowledge". This is nonsensical.

It means (forgive my repeated quotations), a judgement that "rests not merely on experience but on the conditions of the entire possibility of experience which lie within us. For the judgement is determined precisely by that which determines experience itself, that is either by the forms of space and time intuitively perceived by us a priori, or by the law of causality known to us a priori. Examples of such judgments are propositions such as: two straight lines do not enclose a space.--Nothing happens without a cause.--3x7=21.--Matter can neither come into existence or pass way." They transcend knowledge only in the sense that (according to this author) they're truths (like logical axioms) that exist independent of experience, are required for experience, and we have an intuitive understanding of them (on which empirically true statements depend).

Da fuck is going on with my quotes?
Edit: fixed.

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Just because we only know the world by means of the senses does not mean that you cannot reason about the parts of reality that exist apart from the senses.

Good luck with "reasoning" about anything that exists "apart from the senses" - care to explain how this is even possible? Why would you assume that reason even applies to things "apart from the senses" even if there were such things in this category in principle...and how have you determined that there are?

Yet it seems we must assume that certain reasons are unconditionally true when describing the nature of reality as it existed before (and hence, apart) from the senses, in order to give rise to the senses (which, in turn, discovered or gave intelligibility to the reasons). No?