What makes math different from other subjects?
One significant way that maths are different from other subjects is that maths teach abstract concepts using the abstract symbols of numbers.
While it can be said that all language is communicated through the abstract symbols of words and letters, it is also true that these abstract symbols are part of our daily communication experience from the time that we're one-and-a-half or two years old (or even younger if we're read to). It can also be said that, as much as we love maths, we never use the abstract symbols of numbers in our daily communications of trivia, joys, sorrows, loves and irritations. This is because numbers are abstract symbols of a specialized set.
We are all so familiar with the opening gambit of math learning, "2 + 2 is the same as 4," that we forget these abstract symbols are conveying an abstract concept. True, "2 + 2 is the same as 4" originates in a concrete concept, often involving apples (or perhaps pomegranates): "If you have two apples and someone gives you two more apples, then you have four apples." Apples are very concrete: "two apples and two apples is four apples" is a very concrete concept, so concrete that you can eat those four apples. [It is fair to also say that for the literalist, not innately versed in metaphor, "2 + 2 is the same as or is equal to 4" is clearly an untruth for "2 + 2" and "4" are clearly not the same nor equal.]
The abstract concept behind the two and two is four apples is that things manipulated by rules in a non-spatial context, represented by abstract symbols, produce identical results that can affect the spatial world. To illustrate the abstract nature of maths, think of 10 apples raised to an exponent of 20. You do not want a concrete demonstration of the correct calculation for how many apples that results in (as you might want for 2 + 2 = 4 apples). You want to leave those apples on the abstract plane where they can be manipulated by rules. Ten apples raised to an exponential power of 20 yields 100000000000000000000 apples. To reiterate my opening point: maths are different form other subjects because maths teach abstract concepts using the abstract symbols of numbers (in combination with other abstract symbols: + = < ^ - ( ) and so on).
A significant reason math is different from other subjects is because it is more objective and, by its nature, quantifiable. Compare math to, say, English. If a student takes a math exam, typically she can expect to be given a set of problems she is expected to solve for a specific number or value. Almost always, these problems have only one often numeric or symbolic answer. It is thus easy to see if the student is correct or incorrect in her solving of the problem. Subsequently, it’s easy to determine whether the student has correctly learned the skills she needed to solve the problem.
In the case of an English exam, a student is liable to encounter an essay prompt asking him to analyze or interpret text and then render said analysis/interpretation in the form of prose. Let’s say the English exam is based on a class’s recent reading of a book and the questions ask the students to both recollect and interpret what they’ve read. Each student could potentially have a slightly different interpretation of the text while still being considered “correct” in the mind of a professor. Those same answers, however, could be considered “incorrect” in the eyes of a different professor. The process of ingesting, interpreting, and grading prose is far more subjective than is the case with mathematics and other empirical subjects.
Of course, at higher levels, math becomes more and more complicated, sophisticated, and nuanced. Upper level mathematicians sometimes liken math to art, in fact. There is often more than one way to prove an advanced theorem, and then the objective is not just to prove the theorem but to do so elegantly and simply. But most never achieve that relationship to math.
One way in which math is different than other subjects is that each skill or concept builds on other skills. Therefore, if you haven't understood or mastered a previous skill, it will affect your ability to carry out later skills. Think about trying to do algebra if you can't add. This is different than the study of history, for example, in which studying one time period may or may not relate to the study of another time period.
In addition, math has a different language than other subjects. Even foreign languages use alphabets and sentences of some varieties. However, math is denoted, or shown, using a totally different alphabet. Students have to understand this language if they are to carry out even simple mathematical operations, such as adding and dividing. As math becomes more complex, it is even more essential to understand its language and what it means.
Finally, in math, each concept is related in some ways to other concepts. Although it seems hard to imagine, there is a mathematical universe in which all the concepts fit together in some way. For example, the study of geometry and algebra can be related. That is why it's important to use different areas of math together to solve problems rather than seeing these areas as totally discrete, or separate.