It is fairly well known that nowadays derivatives are defined primarily by use of a limit argument. I think I recall that when the idea of derivatives was first introduced (and limits had not yet been used) there was some criticism because the method of computing derivatives/justifying them required the notion of an absolutely small positive number (epsilon).
I am wondering whether the concept of surreal numbers would allow for exactly this definition of derivatives/differentials to be mathematically valid as well. After all, surreal numbers would allow for an epsilon that fits the description above. It would also justify why higher-order terms can be neglected in a derivative (because higher order terms represent differentials raised to even higher powers, which must then belong to even smaller groups of surreal numbers.
Thanks for any help on this.