Hmm. OK. I expected someone to bring up Frege. I'm no expert on his work, so I hoped someone might know more of him than I.
Frege is one of those guys whose importance is demonstrated by the fact that everybody likes to point out why he was wrong. But I have to say, he noted some very cool things.
For example, prior to Frege many considered "number" to be a property. Frege showed how "number" does not meet the criteria of a property. For example, if I have 5 green houses, I can note that "green" is a property of each and every house: "the house is green."
But I can't say, "The house is five." I can't even say, "the houses are five," in the same way as "the houses are green." So, Frege insisted that "number" is not an adjective. It is a noun. And, I think you'll find that most dictionaries list "one", "two", "three", etc. as nouns. That is very curious.
Why?
Well, consider the noun "tree." It can be used as a noun or an adjective. I can walk up a mountain and define a "tree line." The line dividing the place where trees grow and where trees don't grow is of type "tree."
I can't do the same with number. I can't find a "number line" that divides the place where numbers exist from the place where numbers don't exist. So, "number" is not the same as "tree" in that way. Nor is it the same as "tree" in my ability to say, "There is a tree." I can't point out, "There is a five." Five must always be associated with something else, be it five objects, the word (five), or the symbol (5).
There are numerous other examples like that with other nouns.
Yet, Frege seems to be very successful with his definition of number as an object. For example, his definition of "zero" is "the count of things which contradict themself" (or something like that). Pretty cool. Most of all because it embeds the law of non-contradiction in number theory. In other words, the fundamental law of logic also becomes the fundamental law of "number." Just so cool.