What would happen if there was no negative numbers?

Math and our number system, is an invented formal system used to describe real concepts, and has been compared to a language. If we didn't use the concept of negative numbers, that would simply mean that we had coined a different notation to describe everything. All of the laws of nature would be the same, and the universe would be the same, but we would use different terminology to describe certain concepts.

Our system of mathematics might be more awkward, which would cause scientific and engineering advances to suffer. Then again, it could also be logically equivalent and just as good as the one we have now. It is impossible to say which.