The inherent nature of feelings makes them impossible to fully expunge—even from a “cold, rational, logic-based world.” Humanity does a great job of “pushing them aside” or locking them up in boxes (compartmentalization), and of then convincing itself that it has managed to separate the logical from the illogical, thought from feelings, etc.
However, I think there is an ever-present (albeit somewhat delicate, nebulous, ephemeral, and, often, almost imperceptible—sometimes like a gas, at others, more like a vapor) layer of feelings that permeates all thought (“layer” is not the right word because it brings an icing-like image to mind, and I don’t really mean that, maybe thread of feelings?)—a deep and underlying connection between thoughts and feelings that cannot ever be completely severed (or, if it can be, that such a severing is what allows some of the terrible things that occur in this world to occur). A good analogy might be the Peter Panesque idea of trying to separate a shadow from a body. The circumstances (how bright the light of the context in which the entity appears) determine the strength of the shadow—sometimes the presence of feelings is more evident than others (I know, I know, an EXTREMELY imperfect analogy) . . . . but do you get the general idea I am trying to convey here?
I suspect that many people who pride themselves on their rationalism and objectivity and who accomplish much in that realm are probably guided much more by their feelings when they arrive at “objective, rational, logical” conclusions and decisions than they would be comfortable admitting. Most such people would probably agree to the idea of “intuitive, logical leaps” but, in their minds, believe that such leaps are distinct and unrelated to “the emotional” or to “feelings”—which they tend to associate with illogical, even irrational thoughts and behaviors.
I think that because feelings connect what is outside of us with “the essential” that is inside of us, we tend to view them as an Achilles heel. The knowledge that there is such a direct conduit and point of access to our deepest and most fragile vulnerabilities is more than uncomfortable—so we push feelings aside, try to impose boundaries on them, deny their existence, etc. To take that a step further, I think that feelings are more difficult to manage than thoughts because they tend to take forms that are more diffuse than thoughts. Consequently, thoughts seem more “concrete” to us, and “feelings” seem . . . I don’t know . . . incidental? They are certainly messier than thoughts, whose structure and components often seem more clearly defined, and, therefore, more trustworthy.
I suppose I am advocating for the idea that there are many different ways to “know” something, that thinking and feeling are BOTH tools for gaining knowledge, that there are other tools besides just those two (experiencing, for example), and that a skilled craftsman (or craftswoman) understands that there is great power in having a variety of quality implements in his or her toolbox, in knowing which tools are available, in using the appropriate tool for the appropriate task at the appropriate time, and in knowing why one is better than another for accomplishing a specific task.