Davis discusses the problem of gaining information about an infinite sequence from its initial sequence in two cases. In applied mathematics, the question is what information one gains about the limit of a sequence when analyzing an initial sequence. In pure mathematics, the question is what information one gains about the truth of a statement about the whole sequence when analyzing an initial sequence. He then draws a distinction between what he calls constructivism, in which infinite sequences are given by rules for their construction, and existentialism, in which infinite sequences are assumed to exist without further specification of the rules for their construction.
Davis gives nice examples for both applied and pure mathematics. However, he does not defend a clear overall thesis, and his original claims are too unclear or unsupported, or both.
The section entitled "The Paradox Considered" contains what is closest to an actual thesis. But as far as I can tell, Davis only states (i) that the problem described above stems from constructivism but is also expressed in existentialism and (ii) such a transfer is not always problematic. This is no substantial thesis, all the more so because Davis also claims that his distinction between constructivism and existentialism is vague, there are overlaps, and there is ambiguity ("Constructivism vs. Existentialism"). Furthermore, Davis provides no argument for this claim, but simply states it. Finally, I am not sure what use one could get out of this insight, and Davis indicates none.
There are a few other problematic claims and arguments in the article, of which I will only list two:
Davis writes: "[T]he computational strategy of applied mathematics involving iterations is based on the supposition (or the faith) that the particular iterative method employed will converge with sufficient accuracy within an amount of human time that is agreeable to the investigator and be consistent with the phenomenon as experienced." But this does not seem to be the supposition on which one has to rely (by the way, the disparaging moniker 'faith' seems to me uncalled for). Rather, one has to suppose that, given some iterative method, the shut-off criterion is a good indicator of the limit of the sequence. If some iterative method does not converge quickly enough, that is a pity for the mathematician who had hoped that it would, but the mathematician did not need to /suppose/ that it would. By the way, the following paragraph seems to serve no purpose and reads like it it was cobbled together from different texts. I recommend deleting it.
In "Constructivism vs. Existentialism", Davis speaks of the vagueness of the limit definition, but his argument is based on the alleged unclear nature of existential and universal quantifiers and the statement 'given any ε'. These concepts occur everywhere in mathematics and logic, so this does not seem like a problem specific to infinite sequences. Indeed, among the concepts that are used in the sciences, these are probably the most precise ones. And again, it is not clear how this discussion contributes to a solution or clarification of the problem.