There are mountains of evidence supporting evolution and very little to disprove it. Theories in science are not immutable. If there is strong enough evidence to the contrary then any scientist worth his salt will alter his "beliefs".
Also, you seem to be implying that there is a direct link between accepting evolution as fact and their "belief system" (which I assume you are referring to religion), I do not think that is the case.
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There may not be a single piece of evidence that supports evolution but all the evidence combined paints a fairly clear picture.
Things have changed over time.
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One thing that Darwin noticed on his travels, and that people continue to notice today, is that fossils in the bottom layers are very different from the organisms alive today; Darwin didn't even recognize them. As one looks farther up, at younger and younger rock layers, the fossilized plants and animals become more and more familiar until they are a lot like organisms that are around now. The organisms also tend to become more and more complex.
(Carbon dating is not required to know that higher layers are newer)
There are several types of speciation. Speciation has been observed, there is no denying that. But for me the important part of speciation, that supports evolution, is when a new species develops that can reproduce with itself but not the species it came from:
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For example, there were the two new species of American goatsbeards (or salsifies, genus Tragopogon) that sprung into existence in the past century. In the early 1900s, three species of these wildflowers - the western salsify (T. dubius), the meadow salsify (T. pratensis), and the oyster plant (T. porrifolius) - were introduced to the United States from Europe. As their populations expanded, the species interacted, often producing sterile hybrids. But by the 1950s, scientists realized that there were two new variations of goatsbeard growing. While they looked like hybrids, they weren't sterile. They were perfectly capable of reproducing with their own kind but not with any of the original three species - the classic definition of a new species.
Very different species share the same parts.
https://en.wikipedia.org/wiki/Homology_(biology)