Also (minor correction): pi is the ratio of circumference to diameter, not circumference to radius.

But Archimedes's favourite result, which at his request was engraved on his tombstone, was that the ratio of the volume of a sphere to the volume of its circumscribing cylinder is 2:3. Shouldn't that, by your reasoning, be an irrational number?

Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]

I suspect that it's the straight line relating to the curve that will manifest the most analog-digital dissonance.

Again, not necessarily. For example, the arc length of an arch of the cycloid generated by a rolling circle of radius r is 8r.
Cycloid - Wikipedia, the free encyclopedia

A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line. It is an example of a roulette, a curve generated by a curve rolling on another curve.

I will just follow as my knowledge on maths is limited...(I use metaphor and words to replace maths purpose...!) But this is indeed very interesting...please continue and keep it "non-technical" but sustain detail...

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dxArchimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization.

I've had 6 hours of sleep in 3 days.

I think the point partical theory of reality and matter is passe.

It can't be dismissed as passe until somebody has reconciled quantum mechanics with general relativity, because the latter describes space-time as a 4-dimensional Lorentzian manifold, identifying point-events with quadruples of real numbers. (I don't know the exact technical description which would make this statement mathematically precise. Throw in the word "locally", or something! But the point - no pun intended - is that general relativity describes space-time as some kind of continuum, which is composed of point-events, which have to be described, in any given reference frame, by means of real numbers, which are in principle exact. I'm very much open to correction on the details. I never did get around to studying this stuff - which I regret.)

I think the point partical theory of reality and matter is passe.
Along with it the notion that space and time are continuous.
3 of the 4 fundamental forces are quantitized and discontinous.
It seems likely that space and time are also quantized and discontinous.
That fits with the process notion that reality is composed of discrete events "becoming" not "being" and that time is merely the change of process. Transcendental time is a mental construct not a reality of nature?

More broadly, a maddening intuition about the discrete existing within the continuous has been tormenting me for more than thirty years, and I can never get it straight. For me, it is all bound up with equally confused intuitions about gender. (I'm transgendered, or gender-dysphoric, or something.) Masculinity has something to do with the discrete, and femininity with the continuous - of that, I'm fairly sure, but this is a mystical realm in which pretty much everything is paradoxical, and the opposite is almost as likely to be true as well. So, for me, the miracle of the infinitesimal calculus symbolises being both female and male at once - Freud's famous 'phallic woman', perhaps, or (a little more intelligibly to me) the baby as penis, the clitoris as penis, and Ferenczi's concept (in Thalassa, which I didn't get at all far with reading) of the narcissistic basis of the Oedipus Complex.

In my mind math and numbers are a form of logic, logic is a form of reason, and reason (logos) is a product of intelligence (nous).

Reality itself I would maintain is also discrete, digital and quantitized and the impression of continuity in space and time is a product of the conceptual constructs of the human mind. (Much as we can digitize any medium and experience it as continuous). Continuity is a feature of human conception not independent reality. This would make number even more foundational to reality.

Ours ideas of such continuous perfect forms as circles, spheres, etc. can only be represented in mathematical form not in reality. The calculus the discrete representation of the continuous is thus a major breakthrough in the ability to provide a mathematical model of reality.

Well anyway just a sketch of number, reason, quanta, and process and the ways in which they correlate. Is number then transcendent or transcendental? I am not sure how you use the term but in theology transcendental means beyond the material realm, or separate from the material realm and so number I would maintain along with reason exists prior to "creation". Number is a form of truth, of beauty, and a transcendental reality not at all a construction of the human mind imposed on reality but Truth discovered not invented.

This is a very transcendental (if I am thinking of the word from the right perspective) place to approach it from, the concept of a limit being quite odd and transcendental itself. In this way I concur with your assertion that numbers such as PI are dually transcendental, and this is due to the singly transcendental nature of a perfect circle, since it requires infinite density. So when we describe the circumference of a circle divided by it's diameter we implicitly require an infinitely recursive and thus transcendental scheme, and furthermore we are speaking of a relationship between two transcendental schema! So we have reached a second tier of transcendence (I do kind of dislike that term though).

In realizing that we can derive these transcendental numbers from examining the relationship between singly transcendental concepts, we begin looking at appending these new numbers, but also we ask if these 'numbers' are really numbers at all! They are in principle and origin and behavior so very different and contain so much more conceptual baggage than other numbers! I think that Wittgenstein would say that calling these 'numbers' numbers is the result of confusion!
More on this later!

I gather that you are using AND and NOT figuratively? If not, in the literal first order logic meaning they are sufficient. Or can be derived in this way ~(~P^~P). So then we also get implication with P V ~Q. How about neither-nor? Sheffer stroke - Wikipedia, the free encyclopedia

You should read Carnap's Logical Syntax. I unfortunately haven't read it yet, but I have read Quine's analysis of it and it seems very interesting. From what I understand he tries to flesh out and formalize Wittgenstein's picture theory of meaning. That is the most comprehensive reductionist programme I know of when it comes to language. Quine has some very interesting things to say about Carnap's ideas and vice versa (read 'Dear Carnap, Dear Van-the Quine Carnap Correspondence', it's interesting, espceially the lectures at the beginning).