- Do as many steps as you have apples on the table.
- Count or wait as many seconds as you have apples on the table.
- Count is many pencils as you have apples on the table.
- Do as many push ups as you have apples on the table.
You can deal separately with pure number 3, without linkage to any of the objects it can represent the count of. That's pure math. Once you start keeping track what you count, applied math kicks in.
Of course, you are always (as in any scientific discipline) interested to find the truth. Here, you may want to be interested to find truths about numbers. That's where logic enters, with its initial assumptions, axioms, theorems, proofs. You, essentially, always want to prove what is true in math. Mathematicians are after the proofs about counts. Mathematicians are after what is true about numbers, counts and their relations. Lawyers are after the proofs what is true with regards to law, moral, what is right or wrong, and with regards to other human values. Physicists are after the truths in physical world, where various forces, energy, motions are central focus in their investigation. Story writers and movie makers are after the true emotions and true moral messages their work will convey and show, even with fictitious plots, i.e. no matter whether the story is fictional or not, the message about human values, be it emotional, moral, must be real and true, and this message will be true if the story line is logically consistent with the story's framework, no matter how fictional that framework may be.
Now, back to the first example, with apples, steps, seconds, pencils, pushups. Mathematics, while apparently common to all those cases, can not define the actual concepts it has counted. What differentiate an apple from a pushup, and a pushup from a pencil, and a pencil from a second is not part of mathematics, and mathematics is, more or less, not part at all of that analysis and those very important relationships. Moreover, it is these non mathematical relationships that define the various disciplines and it is these non mathematical relationships that very often dictate the direction of mathematical development. These relationships dictate what, when, where, and why will be counted, measured, if required at all.
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