How do finite processes and endless mathematical objects relate to reality?

Question

There seem to be several philosophers who believe science (plus human norms for Sellars) can in principle leave no unanswered questions about reality. I would call this finite or exhaustible.

Sellars:

"In the dimension of describing and explaining the world, science is the measure of all things, of what is that it is, and of what is not that it is not"*

Is describing and explaining the world through science everything to Sellars though? No

"he does not believe that describing and explaining are the only significant dimensions of human activity"

"Sellars would never hold that all our questions are to be settled by scientific investigations."

But the normative nature of the mind, meaning, and language can be reconciled and united with science, elimating all dualism.

"His philosophy of mind abandons many of the assumptions taken for granted by the Cartesian tradition (e.g., the transparent or self-intimating nature of the mental), moving towards a naturalism that respects the normative dimension of mind."

"his own call for a “synoptic vision” that unites a science-generated picture of empirical reality with “the language of community and individual intentions”

"Sellars calls for a stereoscopic vision in which the descriptive resources of the sciences are united with the language of individual and community intentions and the dualism of the manifest and scientific images is transcended."

So in summary: I take Sellars to mean science+norms of language and meaning will transcend all dualisms, giving an exhaustible account of reality. In as much as we understand anything, "it rained" -> "the road is wet"; "the ball is red" -> "the ball is colored", we can understand everything.

Carnap

"Since science in principle can say all that can be said, there is no unanswerable question left. But though there is no theoretical question left, there is still the common human emotional experience, which is sometimes disturbing for special psychological reasons."

I take Carnap to mean, science can answer all questions (unlike Sellars), but the nature of the human psyche may make certain realizations hard to accept (e.g. emotionally, spiritually).

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So to me Sellars and Carnap seem to say an exhaustible understanding of all of reality is possible. What I would like to know (1) is this a fair depiction of their stances (2) how do mathematical objects that suggest inexhaustibility such as, no highest real number, non-computable numbers existing but never "known", Godel incompleteness, halting problem, fit in with the above? The only way I can make them fit is a nominal or fictionalist account, where there really aren't inexhaustible objects. They are limited to and by human activity, e.g. there is the highest real number, the highest one a human ever writes down or uses in a proof.

Answer 1

Short Answer

The questions of the relationships between the tangible and the intangible, the finite and infinite pervade ontological and metaontological discourse. These concepts color discussion in, at least, the philosophy of language, philosophy of mathematics, and philosophy of mind. The question of what is appropriate in ontological commitment (SEP) determines the answer to your question (and 10 ontologists have 11 opinions). How the infinitely abstract and the finitely concrete relate to each other might be considered a corollary to this question, and one of the core philosophical problems of philosophy precisely because it deals with questions most famously (but not exclusively) answered by the famous philosophers such Rene Descartes; this brings to mind mind-body duality, though the issues at hand were first raised by the pre-Socratics.

As such, no single canonical exists to your question (and anyone who disputes it has a flawed ontology. ;)

If one accepts that there are many valid answers because the question of what constitutes truth, an epistemological pursuit, isn't black and white or even necessary in accordance with a bias towards fictionalism, then the question in some regards is a personal question. If one insists with the realists that some correct answer exists, then it is another matter.

Long Answer

Borrowing heavily from Ontology and Metaontology: A Contemporary Guide by Berto and Plebani, we can answer your question in a circumspect way.

Carnapian Ontology

To compare and contrast your fomulation of Carnapian ontology would be a lengthy process. A good free start is Metaontology (Oxford Handbook). Chapter 5 in Berto and Plebani is also devoted to Carnap. A brief summary infused with some common knowledge here suffices.

Rudolf Carnap was a logical positivist and held a certain attitude to metaphysics as was the case with those positivists and empiricists. He was a teacher to Quine. He held in essence that such questions could be internal or external where internal questions about existence were essentially ordinary technical and scientific sorts. 'Does a banana exist on a table in a room' is a question a scientist is poised to easily tackle by logico-empirical methods. External questions were the metaphysical sort, such as do numbers exist generally? Quine attacked the internal-external distinction by attacking the analytic-synethetic distinction. Quine is generally considered to have the stronger argument, but not everyone agrees.

For Carnap, meaningful questions exist in a syntactical linguistic framework and one merely appeals to that framework for determination of the truth condition, ontological questions not immune. Since terms are operationalized, one merely appeals to (what is now called) the public language and the results of operational methods, and the truth of the ontological claim is determined. Any question which flirts with using (sometimes deliberately) obtuse private language and/or avoids operationalizing concrete and abstract things is meaningless (Unsinn by way of Frege). Thus science, and its siblings of math, technology, and engineering are enough to address all things real. In this view, Carnapian thinking deals with mathematical infinity by seeing it more as a linguistic artifact since it doesn't neatly fit in with the received view of spatially extensive, punctilinear time. In other words, if Minkowski space-time, or some similar framework doesn't account for it, then the question of "existence" is rather meaningless in a physical sense.

(I simply am not familiar enough with Sellars's work to explicate.)

Platonism and Fictionalism

Simply put, Platonic thinking is the idea that Forms as abstractions are more real than physical reality, the latter of which he characterized as ephemeral and diseased. Fictionalists essentially see the question of the "one true reality" as somewhat moot, not because they deny a shared, common physical experience, but because they argue that to attribute an absolutist truth-conditional semantics to a language overestimates the purpose and essence of a language; a language is a tool where the pragmatic impact of the language is what is of value. In both cases, there is no problem with the phrase "the infinite exists" simply because it is either a linguistic utility or some form of idealist notion of ontological commitment.

Summary

You asked:

How do mathematical objects that suggest inexhaustibility such as, no highest real number, non-computable numbers existing but never "known", Godel incompleteness, halting problem, fit in with [reality].

The short reply is it depends a lot on your presumptions about existence, your ontological commitments, as it were. Too many positions exist for a pat answer; for Quine, the very act of enumeration determined existence. For Carnap, it was about empirical and logical operationalization. For others, such as theologians, it is fiat, the product of a transcendent Creator. Differential ontologists appeal to immanence and the paradox of time to make claims about existence. My personal take is to appeal to an embodied cognition within an evolutionary-minded naturalist physical universe. Infinity is merely a linguistic abstraction to describe a fictive upward bound on linear processes for pragmatic purposes. This allows me to claim that human thought is simply a superset of the Turing machine and all the information processing cycle inhabits and appeals to the philosophy of information.

So, like many questions philosophical, you'll have to assemble your own satisfaction. Good luck!