Mathematics is objective in so far as the defined axioms correspond
to reality. The axioms themselves are completely subjective
assumptions as they can't be proved.
If we agree on objective laws which cannot be proven to be
unconditionally and totally true, we can treat those laws as
self-evident in a system.
The most prominent example to start with is
Peano's Postulates:
[
Guiseppe Peano(1858-1932) was an Italian mathematician who devised
a set of axioms that can be used to prove the existence of natural
numbers and all of its usual properties. These are known as the
Peano Postulates.
Postulate 1: There is a number called 1.
Postulate 2: Every number has a successor, a natural number that
comes after it.
Postulate 3: The number 1 is not the successor of any natural number.
Postulate 4: If two natural numbers are different, their successors
are different.
Postulate 5: If we have a set A of natural numbers that contains 1
and also contains the successor of all its elements, then A is all
the natural numbers.
Postulate 1 says that the number 1 exists. Postulate 2 says that
there is a successor to this number, usually written as 1'. Postulate
3 says that 1' is different to 1. We call this number two,
and write 2 = 1'.
]
1 + 1 = 2 is analytical and therefore self-evident. 2 is the same
thing as 1 and 1 put together and vis versa. Question the validity
of this is asking how do we know 1? which is the same thing as
asking how do we know something exists? You're questioning reality
itself. These are one the axioms believed to allow us to move
forward.
I think if I were asked why dos 1+1=2, I would start by pointing
out that, in both instances of (1) + (1) = 2, here we have 1 being
accepted in its definitiveness of value/quantity existence.
It therefore meets all rational and logic that when its
definitiveness of value/quantity existence is increased
by its similarity the definitiveness of value/quantity
existence clearly cannot be the same. This definitiveness
no longer enjoys a singular value/quantity existence.
This then forces us to accept a new definitiveness of
value/quantity existence , i.e. 2
This principle of value/quantity definitiveness applies to all
numbers. I say numbers, because I don’t accept zero [0] as a
number since it has no definitiveness value/quantity existence.
