do negative numbers exist, or are they merely abstractions?

this one is a little more tricky, and, while it's hard to tell given my age at the time, i think i may have lost my debate partner, who, unlike myself, had finished the second grade - along with the second year of university.

it was in the years between when my parents split up and my mom remarried, and i was going to school full time, so i guess it would have been in the first or second grade. my mom got a room mate over those years, who was a philosophy student at ottawa university. i was always told the relationship was platonic, but i always thought otherwise. she officially slept on the couch.

the most important thing i remember about this woman was that i was to never, under any circumstances, touch her organic peanut butter, which was very expensive. i could have as much of the regular peanut butter as i wanted.

she let me try it once, and i didn't like it anyways. chalky. dry. sugarless.

they told her i was smart, and she wanted to test for herself, so she asked me to come over and talk about things. and, i remember that we talked about a lot of things, but the only part of what we talked about that i remember was negative numbers.

i was quick to react.

"there's no such thing as negative numbers."

she frowned. but, i continued.

"a number is the same thing as how many things there are..."

..and, she shifted in her seat a little at that...

"....so, what the number 3 means is that there are 3 things. how do you have minus three things?"

the frown lifted a little, as she became aware that i was thinking in what she understood as the correct terms.

well, she said, you have minus three things when you take away three things.

i rejected this, flatly. for, taking away three things is not the same thing as having three negative things - that is not existence, that is merely a way to describe things that don't exist.

and, then she did something that neither of us were expecting to do - she pulled out a textbook and explained to me that there was an idea called natural numbers (the whole numbers you can count), which were a subset of real numbers. i remember understanding this, or at least recognizing it as an obvious truth. then, she scrawled this backwards E on a piece of paper, which i did not understand the meaning of.

because negative natural numbers are a subset of real numbers, that means they must be real, that they must exist.

what i remember from this point was being confused. i trusted her on some level; or, at least, i knew she ought to be right. but, it didn't sit right with me. how could negative numbers exist? how could an absence of something be a real thing?

but, i had no argument.

i know today that her argument was wrong, and i maintain my own intuition on the matter, but neither really deduces the correct outcome.

so, what do you think?

do you think they exist?

jagmeet singh must cut his beard