An inferential chain is a series of inferences where each depends on the previous in sequence. "From A we conclude B, from B we conclude C, and from C we conclude D." That would be a chain of length three, as it has three inferences.
In contrast, "A supports D, B supports D, and C also supports D" is not a length 3 inferential chain, because the order of the three inferences can be rearranged without altering the argument. The longest inferential chain in that argument is only length 1.
To be more specific, we can consider an argument as a directed acyclic graph from premises to intermediate conclusions towards final conclusions, with an arrow from each justifying statement pointing towards the statement it justifies. An inferential chain is a directed path in this graph.
In mathematics we do have long inferential chains. Mathematical proofs may run into the hundreds of pages, and the longest inferential chain through such a proof would also be very long. This is possible because each step of mathematical reasoning is very certain. So we can put many links in the chain with a low risk of it breaking. (Still, very long and complicated mathematical proofs are not infrequently found to have flaws.)
In ordinary common-sense reasoning, inferential chains tend to be much shorter. "It's raining today, so I'll bring an umbrella," has only length 1. One good explanation for why common-sense inferential chains are short, is that common-sense inferences are uncertain. If each inference is only 90% sure, then if you chain 10 of them together, the chain is only 0.9^10 = 35% likely to be intact without broken links. So in ordinary common-sense reasoning we prefer to build broad, rather than deep, arguments. We prefer to consider the pros and cons that add or subtract directly from a claim, rather than extrapolating too deeply in too many steps from the available evidence.
In fact, often, common-sense arguments are neither deep nor broad; people find it difficult to subjectively weigh too many pros and cons, especially once there are more pros and cons than will fit in short-term memory. And so we often fall back on picking out one or a few pieces of evidence as most important, allowing ourselves to ignore or override lesser evidence.
How can we characterize the inferential chains used in philosophy? Are they long, like mathematics, or short, like common-sense reasoning?
It's my feeling that legitimate inferential chains in philosophy are short. Because in philosophy (except for logic) we don't have the luxury of reasoning with perfect certainty, and rather must appeal to intuition and analogy most of the time, long inferential chains have a high chance of being flawed. The situation is more like common-sense reasoning than mathematics.
Philosophical papers are often very long. But famous specific arguments in philosophy can usually be summarized within just a few paragraphs. Without the luxury of logical certainty, an excessively long argument becomes unwieldy and unsure, and we are unable to fit the whole thing in our heads or reason effectively about it. The length of philosophical papers often comes from comparisons to other works and examples or hypotheticals, rather than from having a long central argument. Is this perception of the state of affairs correct?