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There is no reason that you cannot see your son without (i.e., before) the family court issues a formal ruling/schedule on visitation. The fact that you were not married to his mother does not even matter.
Of course if there is a restraining order or order of protection against you then that would prevent visitation for now.
Is the mother giving you problems about visiting with your son?
A colleague of mine just reminded me that as you and the mother were not married set can set limitations on your visitations (including not allowing unsupervised visits) until paternity has been established (usually through laboratory testing of blood or saliva.)
So until this is accomplished you should agree to her limitations. In the meantime, file the paperwork with the local family court to establish paternity and get custody and visitation issues resolved. Also, remember that child support will likely be ordered as a part of these proceedings.
Unfortunately a birth certificate is not helpful.
Paternity needs to be established formally and legally. Birth certificates are produced more often than most people realize where the name of the father is not the biological father, so a birth certificate cannot be used to legally establish paternity.
You should, however, be able to arrange for a blood test to establish paternity with the court before the hearings take place. The sooner you file the necessary paperwork the sooner you will have your first hearing and be able to establish paternity and request a temporary visitation order.
Over time the rest of the details will be worked out as things never move quickly in courts.