In most cases the numbers that normal humans use don't follow strictly the axioms of Peano. Most of the time the dates of a month follow Peano. Most of the time the day after September 2 is September 3. But not always.
On a computer you can't store every natural number as specified by Peano with common integers.
If you start counting apples and get really many apples you suddenly have a black hole and no apples anymore.
I don't know enough about the philosophy of math to get really deep by we had lately someone writing posts about constructivist math that also contained the notion that there are no absolute mathematical truths.
In that case, it sounds like you're just not a math realist. There are plenty of people who believe that Peano arithmetic somehow exists on its own. Or possibly people who have a different definition of "exist" from me. It's hard to tell the difference. But I don't think disagreeing with that is all that unusual.
As per a recent comment this thread is meant to voice contrarian opinions, that is anything this community tends not to agree with. Thus I ask you to post your contrarian views and upvote anything you do not agree with based on personal beliefs. Spam and trolling still needs to be downvoted.