Then why do they give dates in the thousands at all? What is the relevance of the dates given?

it's my guess that the "greater than" symbol means exactly that.

example >53,000 years:

the above means the date is greater than 53,000 years, the sample can range from 53,000 years to infinite

Yep that's right.

http://www.c14dating...adiocarbon ages

*"If the sample approaches D14C = -1000 per mille within 2 standard deviations, it is considered to be indistinguishable from the laboratory background, ie, not able to be separated with confidence from the laboratory countrates which result from a sample which contains no radionuclide. In this instance, a minimum age is calculated. An example of a minimum age is >55, 000 yr or >50,000 yr (Gupta and Polach, 1985).... Samples whose age falls between modern and background and are given finite ages."*

It seems to me they are saying between 9,670 plus or minus 60 years, "to" 53,170 years.

If I accept your code it would translate thus; "9,670 plus or minus to infinity/greater than 53,170"

The word "to" would make no sense, what is the significance of putting, "to". It seems like the estimate is meant to be 9,670 years to 53,170 years.

Here's the part you are referring to again.

*“In Eurasia three hundred and sixty RC dates are tabulated from many sources based upon C-14 dating of bones [~130], tusks & molars [~190] and soft tissue [~40] by Yurij Vasil'chuk et al.9 On the Eurasian continent the dates ranged from 9,670 ± 60 to >53,170 RC years BP.*”

You somehow seem to think they are talking about a single sample. Clearly they are referring to many samples (360) from different published sources yielding a range of dates, with the oldest minimum date at >53,170 RC years BP (an infinite date)