I dispute that mathematics or logic can be argued not to exist - they exist by definition. By which I mean, by the time you’ve got done defining the terms of the statement “2 + 2 = 5”, you then know what “2” is, what “+” is, what “=” is, and what “5” is, and based on the definitions you have applied to those terms/symbols themselves, you know that “2+2=5” is false. Similarly, logic also stands on its own definitions.
Now, of course you could argue that perhaps you define “5” to mean what normal people mean when they say “4” - but in that case, you’re doing the same math as everyone, just with different symbols. It’ll make communicating with anybody on the subject crazy-difficult, but it doesn’t actually mean that your math is any different. “II + II = IV” is “2+2=4”, mathematically speaking; what symbols you are actually using is irrelevent.
As for everything else in your senses, memory, or imagination - that’s all of doubtable veracity. Your memories and sensory inputs could be being faked, or could (in theory) actually be completely random; in fact, you may have no past or senses at all, but merely be existing for a fractional moment, with the memory of all your past and current senses and experiences having just fallen into place for this instant, ceasing to exist an instant later. Or you could be more permanently existing, but frozen in time with artificial memories and senses; characters in a book or movie are this way on any given page or in any given still image.
The only reason we believe anything we remember or sense is because it’s really darn consistent. I don’t know if the door is actually real, but I do know that experience tells me that if I try to impell the action of walking into the door with the body I appear to have, that apparently-real body is likely to give me something realistically approximating an unpleasant sensation where it whams into the apparently-existing obstacle. I trust the past experience as consistently predicting the future results of my intended actions, so I don’t walk into the door.