Do I have to give my Social Security Number on my union application?

I am required to join my local union for my job in New York State. On the application to join the union, they ask for my Social Security Number, but being concerned about identity theft, I do not supply this number to those who ask unless required to by law. As I understand, legally, they do not require it. Does anyone know for certain?

http://www.privacyrights.org/fs/fs10-ssn...
http://www.ago.state.co.us/idtheft/ssn.c...

I know that union dues are tax deductible, but so is a donation to a charity. I don't need to give charities my SSN. My pension is setup through my employer's HR department. My employer deducts the union dues and gives it to my union.

no

edit: you do not need your ssn to access your benefits. they will assign you account numbers for that if you choose not to give them your ssn. the only things you must give your ssn for is for your employer to manage witholdings and to collect government benefits. most retirement benefits are tax deferred. you will not need to pay taxes, and therefore give your ssn, until you collect.
any person who denies you services because you did not share your ssn is subject $1000 punitive damages in civil court.

Yes, they have a legitimate need for it - all your benefits and pension accounts.

Do you want to be included in retirement and pension benefits? They need this number to set up accounts.