Anonymity can be absolute or relative. It is important to understand the difference. In the privacy business especially, things advertised as "anonymous", mostly offer only relative anonymity, while the customer can be misled (not necessarily deliberately) to believe the anonymity to be absolute.
Absolute anonymity is anonymity in all directions.
Relative anonymity is an anonymity covering some directions, but not all.
Let us illustrate that with an example. To make it simple, we choose one having only two directions for the anonymity to cover.
If a bank offers an anonymous account (they did in some jurisdictions, in a not too distant past), it is an account without a name. Transactions to and from the account can be made without letting the other party of the transaction see who holds the account. The account is anonymous - in relation to other parties of the transaction.
But how about the bank? If they require to know who you are to allow you to open the account, then it is not anonymous in that direction, that is, in relation to the bank. Thus the anonymity of the account is not absolute, only relative. In order for it to be absolute, the bank must not require a name to open this account; they must stay ignorant of who is the account holder. (Even such accounts existed until quite recently in Europe, for instance in Austria.)
It is worth mentioning that "anonymity" means "no name", and openly shows no name. Often it is confused with the much more common "pseudonymity", which provides a name, but one which is especially created for a certain occasion or purpose.
Yet, strictly speaking, no bank account can ever be anonymous; it is always pseudonymous, because a so-called anonymous account has a number serving as a name. Even if you don't know who owns the account, it is possible to link various transactions done to and from it. They are connected by a common identifier; there is connectivity.
Indeed, genuine anonymity is rare.
So let us define this more exactly: something that is anonymous is not attached to a name, while something pseudonymous is normally attached to an assumed name. This definition can be misleading, because a name doesn't have to be a name in a conventional sense. It can be any identifier, also a number. If that identifier is used more than once, we have gone from anonymity to pseudonymity.
Because of connectivity. We have two events that are connected by a common identifier. Genuinely anonymous events are isolated, that is, they cannot be connected to any other event. For an investigator, connected events offer more clues. The more events that are connected, the harder it is too cheat a snooper. It is hard, however, even for a professional investigator, to find out conclusive facts about an isolated, unconnected event. Qualified investigation is based much on finding connections and by them gradually build up a full picture of something.
From here, it is a small step to the often repeated advice never to use a Bitcoin address twice. If you do, you create an unnecessary connectivity (between transactions), which is detrimental to your privacy. If you really desire privacy, you must minimize all connectivity, everywhere. I would say eliminate it, but that can be practically impossible. Yet, avoid connectivity where it is possible to avoid it. Always using a fresh Bitcoin address prevents undesirable connectivity between your transactions.
Here it should be noted that, while connectivity is detrimental to your privacy, pseudonymity is not always failed anonymity. Sometimes the users of pseudonyms want connectivity. Apart from privacy, there can be other reasons to use a pseudonym.
One we can see in my article “The Toyokuni Confusion”. The way these Japanese artists use their self-assumed names is a means to mark the continuity of a tradition. In this case an artistic tradition. It is exactly the same as kings and popes, who assume a name to mark their place in a tradition. In a way, the name of a king or pope can be seen as a pseudonym. Louis XIV and Toyokuni III are applications of the very same idea and principle.
This is not limited to kings and history though. Let's look at Bob Dylan. I think most of you know who he is. But it is not his “real” name, that is Robert Zimmerman. It is publicly known, so obviously he is not using a pseudonym for privacy, but for two other reasons.
The first is to create a trademark for his activities as an artist. “Bob Dylan” is catchier than “Robert Zimmerman”; and many pseudonyms for artists have the character of a trademark. However, he has also stated himself somewhere, that he chose “Dylan” after American poet Dylan Thomas. The use of “Dylan” indicates an artistic/spiritual connection to the poet, a sort of tradition, even if implicit and indirect. It is not the only case when an artist by choice of pseudonym wants to mark his/her artistic, intellectual or spiritual connection to someone they feel has had an impact on their own artistic or intellectual development.
The fourth reason for using a pseudonym is simply to compartmentalise life. If you are active within different areas of art or business of any description, you might want to keep them strictly separate and for that reason choose to practise them under different names.
So we end up with four possible reasons for using a pseudonym – and there might be more. They are not mutually excluding one another; overlapping is possible.
For the three latter ones, connectivity is necessary, but it always remains a weakness for privacy.
Related article: Why Should You Always Write under a Pseudonym?
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