For example, Euclid's first axiom is that it is possible "to draw a straight line from any point to any point." We have a hard time imagining two points that cannot be connected by a straight line, so we grant this assumption and he moves on with a bunch of proofs about triangles. But he could not proceed with his proofs if you do not grant him that initial toehold.
When Sir Isaac Newton formulated the physical laws of motion, one of his chief axioms (which he probably never thought to question) was that time moved forward at a rate of one second per second, regardless of your frame of reference.
After 200 years, Einstein proved Newton's axiom was flawed. It turns out time flows slower for moving clocks than for stationary clocks. Space and time are intimately related, but none of us had ever felt that before because none of us had moved so fast as to observe it. (We still haven't moved fast enough to feel it, but the effects have been measured by very precise clocks).
In mathematics, axioms are treated very thoughtfully and documented well. In other disciplines, the underlying assumptions are very often invisible to everyone, including the arguer. This can have treacherous effects.
Star Trek made hay out of the apparent disparity between logic and emotion (Mr. Spock vs. Capt. Kirk, Mr. Data vs. Counselor Troy), but actually each worldview is both logical and emotional. They just have very different axioms guiding them.
Our society, of late, emphasizes mathematics and the hard sciences: the technical "logical" worldview. Quite often in my physics classes, my professor would begin by describing a scenario, and in order to fit the problem to the set of equations he was teaching, would make "simplifying assumptions."
"If we assume the cow is a frictionless sphere..."
"Imagine this one-inch thick plate is infinitely long and wide..."
"We assume that all pulleys are mass-less and frictionless..."
"Consider a circular pipe of infinite radius..."
While these were very helpful in uncomplexifying my homework, I believe it also had a subtle effect on my worldview. The unstated axiom that seemed to seep in is that all things are reducible to sterile abstractions. And the corollary, if every thing is a sterile abstraction, each thing has equal value.
"Trees are merely particular configurations of protons, neutrons, and electrons like you or me or that mountain."
"Grandpa's bedsores are fascinating epidermic reactions to sustained friction."
"There is no logical reason to cover our nudity in public."
In contrast, the human-centric "emotional" worldview accepts the following assumptions: people are real and people are more important than things. It also takes into account the entire, complex human system, an obstacle the technical worldview tends to discard.
The human-centric worldview is just as logical as the technical. For instance, there is a very logical reason to eschew public nudity, given their axioms: seeing another naked human has real psychological and physiological effects on the human system that may be difficult for some people to manage. (Granted, the words "psychological" and "physiological" come from the technical domain, and these technical fields are catching up to the human-centric worldview in many important ways, but from an incompatible axiom.)
Similarly, in the debate on curse words, the technical world cannot see any qualitative differences between the words "sass" and "ass," since they are merely composed of a rearrangement of the same two phonemes. That is perfectly true in a simplified universe populated with no humans, but as soon as you collide those words with a human (English-speaking) ear, you will find a qualitative difference in her reaction. And because people are important, we choose which word we use with care.
Emotion is an integral part of the human system and it is also perfectly logical within the human-centric worldview. It is through emotion that human needs and wants are communicated, comfort is given, stress is managed, and decisions are made. Emotion only appears illogical to those who apply simplifying assumptions to the human system.
Because human systems rely on emotional signalling to place weight on ideas, and because humans comprise the majority of those who think technically, the technical worldview is very highly emotional. Think of the sense of pride you get when you solve a difficult problem, the disappointment and frustration that follows checking the answers in the back of the book. The superiority you feel when you leave a comment on an internet message board. We humans are driven by emotion, whether we accept that fact in our worldview or not.
Having a technical outlook is good and helpful for certain situations. So is a human-centric point of view for many more situations. There are myriad other ways of viewing the world, too. The important thing is to recognize your underlying axioms and take stock of how they may be influencing your arguments and decisions. And understand how someone else might choose their axioms.