Arithmetic and the study of arithmetic have been around for many centuries. Used by people to trade with each other, understand each others' problems, build houses etc. Arithmetic is a huge part of everyday life for everyone on the planet. So why do we have arithmetical ideas and concepts? I think this is pretty simple. Arithmetic exists because we need it to live and interact with each other. A good way for us to understand each other is through arithmetic. Although it sounds like arithmetic was found by humans, there is no way that it could have been created by us. Arithmetic is more of something that was discovered, although it already existed in the world around us. It was discovered so we can use it to figure out everyday problems and to understand the people and world around us. Later through extensive mathematics arithmetic has also become commonly used in high level mathematics where things may not relate to real life right now or sometimes never.
It is crucial to understand the difference between two kinds of mathematics to really understand the question of arithmetical truths being empirically falsifiable or not. These two contexts in which we can analyze mathematics are pure mathematics (imaginary world) and applied mathematics (the real world around us). The imaginary world is the world that is created by formulas and mathematicians to try to understand the world in a general matter with certain theories while applied mathematics deals with real world problems rather than going for a general explanation. We can make this distinction by saying that pure mathematics never really only deals with the real world when it is applied thus causing it to be used as applied mathematics. Thus pure mathematics to a point is the cause for applied mathematics but this does not mean that pure mathematics deals with real world problems but rather might be the answer to some of the problems in the real world.
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