Just wanted to add a little extra context to ます and it's modern day 否定形, ません. ます appears in the Nara period as an honorific way to say be/go. It was later used as a honorary supplementary verb (a verb attached to the end of another verb to show respect/honor). The latter usage survives into the modern day with the same usage.
The difference between classical and the modern day is that the classical verb was a 四段 verb so the 未然形 was まさ and the simple negative was まさず. Eventually the 連体形 of ず (simple negative auxiliary) as well as all verbs/adjectives/adjectival verbs replaced the 終止形 (modern day 辞書形). So まさず become まさぬ. Eventually ぬ (negative auxiliary 連体形／終止形) became ん in 関西 (Kansai region) and ない in 関東 (Tokyo and surrounding areas).
So you would expect like all 五段 verbs that ます would've been conjugated in modern Japanese like まさない or まさん but this clearly isn't the case. Instead it is conjugated like a サ変 verb + ん(ぬー＞ん), ません. This is more than likely due to the misinterpretation of ます as being the combination of (何々 "something")＋す. This is completely understandable as a lot of verbs in classical/modern Japanese are formed by something + す. For example, to make a Sino-Japanese noun a verb just attach す (or in modern Japanese する). All of the verbs formed this way follow a サ変 conjugation because they are inflections of す／する.
In short. ます exists as a verb with the same function in both classical and modern Japanese. At some point, ます was misinterpreted as a サ変 verb and is now conjugated like a サ変 verb plus ん.