Why Mainstream Science is a Religion

There is a lot of discussion about this issue on this internet, so I think this question may be worth addressing seriously. The main point of the debate seems to be the following:

Over the past decades, several research groups of self-proclaimed creationist scientists have claimed discoveries of dinosaur bones that they have managed to date, using radiocarbon dating methods, at some age which is a lot below the 'usual' i.e. mainstream accepted date for the age of these bones (several dozens of million years old). The age that these groups claim to find is usually on the order of thousands or tens of thousands of years old. The particular example you bring up is one of the most famous such cases. The claims are really quite spectacular, when taken at face value, and therefore should be examined thoroughly. In this answer, I will try to go through this story in great detail, (hopefully) exposing the reasons why this work is not taken seriously by scientists.

The research by Miller et al.

A research team from the CRSEF, or Creation Research, Science Education Foundation, led by Hugh Miller, has claimed to have dated dinosaur bones using radiocarbon methods, determining them to be no older than several dozens of thousands of years old. Let's look at their research methodology in detail (indicated by bullet points):

As it turns out, Miller's research group obtained their sample in quite a remarkable way. In fact, the creationist posed as chemists in order to secure a number of fragments of fossilized dinosaur bone from a museum of natural history, misrepresenting their own research in the process of doing so.
When the museum provided the bone fragments, they emphasized that they had been heavily contaminated with "shellac" and other chemical preservatives. Miller and his group accepted the samples and reassured the museum that such containments would not be problematic for the analysis at hand. They then sent it to a laboratory run by the University of Arizona, where radiocarbon dating could be carried out. To get the scientists to consider their sample, the researchers once again pretended to be interested in the dating for general chemical analysis purposes, misrepresenting their research.
Let's take a little pause to consider the general issue of misrepresenting your own research. It is understandable that Miller et al. did this, since there would have been a slim chance (at best) of the museum curator providing them with any dinosaur bone fragments if he or she had known what the true intent of the supposed chemists was. In particular, it is implausible that it would have been considered worthwhile to try to use radiocarbon dating methods on these bones, since the rocks that they were taken from were determined to be 99+ million years old, as shown in this paper by Kowallis et al. Now, it is known that 14C14C decays at a fast enough rate (half-life ~6000 years) for this dating method to be absolutely useless on such samples. Thus, it appears that Miller et al. would not have been able to obtain this sample, had they been honest about their intent. This, of course, raises some ethical questions, but let's brush these aside for now. We proceed with the examination of the research done by Miller and his fellow researchers from the CRSEF.

What exactly are we dating here? Sample contamination and general trustworthyness

After the samples were submitted by the laboratory, Miller et al. were informed by a professor from the University of Arizona that the samples were heavily contaminated, and that no collagen (where most of the carbon for 14C14C dating comes from) was present. Miller let assured the professor that the analysis was still of interest to the group. The issue of contaminations is quite a serious one, as can be seen in this paper by Hedges and Gowlett (sorry, paywalled!!!). I quote (quote also reproduced in the paper by Lepper that I linked earlier:
At a horizon of 40,000 years the amount of carbon 14 in a bone or a piece of charcoal can be truly minute: such a specimen may contain only a few thousand 14C atoms. Consequently equally small quantities of modern carbon can severely skew the measurements. Contamination of this kind amounting to 1 percent of the carbon in a sample 25,000 years old would make it appear to be about 1,500 years younger than its actual age. Such contamination would, however, reduce the apparent age of a 60,000-year-old object by almost 50 percent. Clearly proper sample decontamination procedures are of particular importance in the dating of very old artifacts
It is clear that the sample provided by Miller did not under go any 'sample decontamination procedures' at all, and it is therefore strongly questionable to which extent it can be used to obtain a good estimate of the age of the bones. Furthermore, it appears less than certain that the carbon found in the bones actually had anything to do with them being dinosaur bones. In the article by Leppert, we find:

Hugh Miller generously provided me with a copy of the elemental analysis of one of their dinosaur fossils. Daniel Fisher of the University of Michigan’s Museum of Paleontology examined these results and concludes that there is nothing whatsoever extraordinary about them. The predominant suite of elements present and their relative percentages (including the 3.4% carbon!) are about what one would expect to find in hydroxyapatite and calcite, two of the commonest minerals present in ordinary dinosaur fossils. There is absolutely nothing unusual about these fossils and no reason to think the carbon contained in them is organic carbon derived from the original dinosaur bone.

Robert Kalin senior research specialist at the University of Arizona’s radiocarbon dating laboratory, performed a standard independent analysis of the specimens submitted by Hugh Miller and concluded that the samples identified as “bones” did not contain any collagen. They were, in fact, not bone.
These results corroborated established paleontological theories that assert that these fossiles presumably were 'washed away' over long periods of time by ground water, replacing the original bones with other substances such as the minerals naturally present in the water, implying that this sample could not tell you anything about when a dinosaur lived (or rather, died).

Conclusions

At this point, it is quite clear that there is little reason to trust the research by Miller's research group. In fact, the article by Leppert raises a number of additional issues (e.g. Miller's group refuses to reveal where some other samples of theirs were dated), but I think it is pointless to argue further: It is obvious that the CRSEF research group did a poor job in sticking to the scientific method, and that little objective value can be assigned to their supposed findings.