Mathematics is a bit like music, most obviously in its abstract structure, though considerations of beauty and elegance are also important to many participants in mathematical activities. Originally concerned with the ordinary human activities of counting, measuring and comparing, mathematicians have for thousands of years investigated a vast, mysterious world of abstract objects such as the many types of numbers, though these by no means exhaust the mathematical world. There are discoverable principles and processes relating these strange objects to each other. These too are part of this abstract, but very real, world.
Of course, mathematics has continued to be crucially important in counting, measuring and comparing, especially in science, engineering, economics, statistics and all aspects of the modern world, including information technology. In all these areas, mathematicians have been vastly successful in developing mathematical models which model curious aspects of more concrete reality.
Nevertheless, mathematics is not dependent on this physical word, but have a reality and validity of their own, depending on clarity, consistency and completeness. It is indeed amazing that bizarre mathematical objects and their equally curious relationships have been found to model the behaviour of, say the fundamental particles of quantum mechanics. Nonetheless, the investigation and discovery of the maths often pre-dated, and are therefore independent of the physics. It shouldn’t have to be said that physics doesn’t deal with or exhaust all reality. There may be a dependence but it is not clear what the nature or direction of this relationship is.
Logic is another abstract realm, not obviously dependent on other aspects of reality. Logic is historically based on figuring out how one statement implies or excludes another. It was once thought it would possible to show that all of maths, or at least number theory, could be derived from logic, ultimately the law of the excluded middle. This says it is not possible to hold that X and not-X are both true. Of course, you can add qualifications to the X on either side to make this seem plausible, but in the bald case, if you say X and not-X at the same time, the result is you are saying nothing. This logical enterprise was not successful but a whole new world of formal, or mathematical, logic was opened up, much of it forming the basis of information theory. Interestingly, progress in IT has often been by way of finding effective and efficient physical models of the abstract findings of mathematical logic - the opposite of what has been the case with physics.
Mathematics and logic, therefore involve grappling with an abstract realm reflecting upon and related to, but not dependent upon, activities of counting, measuring, comparing, reasoning, etc within a rich set of well-grounded traditions passed on in schools, universities, professional bodies professions jobs and enthusiasts.
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