Waiting for atpollard (and others) to prove the Greek word "adelphos" MUST mean "shares the same biological mother."
Many words, both in Greek, English, and many languages have multiple meanings. Can you prove "adelphos" DOESN'T mean "shares the same biological mother". No you can't.
All we can do is look at the context of who is talking, who is being talked about, and what is being referenced to form a "most logical" conclusion of the correct meaning of a word that can have multiple meanings. Short of having the person who wrote it down in front of you to ask which meaning they meant for it to be, there is no "proof" one way or the other. All we can have are inferences based on context and normal use cases.
You can usually tell from context if someone is calling someone "brother" in a figurative way. Even in modern English, if we have a particularly close friendship with someone we may call them "Brother". Or if they are a fellow believer in Christ we may call in each other "Brother" and "Sister" (which was the practice of my rural Baptist upbringing).
Now if you take a look at where the New Testament refers to Jesus's "Siblings". It almost always refers to a close familial relationship.
Take for instance Mark 6:3 I
s this not the carpenter, the Son of Mary, and the brother of James, Joses, Judas, and Simon? And are not his sisters here with us?' And they were offended at him.
In this context Jesus is being ridiculed in His hometown. They are basically saying "Who does he think he is?" "He isn't any better than the rest of us". He is a carpenter, His mom is Mary, and we know his brother and sisters.
I believe the text says "Brothers" and means "Brothers". The usage seems to indicate brothers and there is a better Greek word for "cousins" or "relatives" (συγγενής or suggenes) that would the writers would have used to indicate someone not of immediate family.
That leaves the possibility that James, Joses, Judas, Simon and the sisters are step brothers and sisters. Which is a more reasonable explanation if you believe in the perpetual virginity.
However, one of the issues with "tradition" in this regard is that there is no set tradition on who James and the rest are. Some say that they were step-siblings from an earlier marriage of Joseph. Some claim they are cousins, which was popularized in the 4th Century by Jerome.
Traditions tells us who they were not (Children of Mary) but it doesn't tell us who they are. I think I'll just go with the Bible and call them Brothers and Sisters.