Mathematics

If we define the science as the study of reality to find the relations among objects through scientific method, is mathematics a science? Or, it's something like logic which can be used in a wide range of sciences, philosophies, and even daily usages.

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I don't much like the expression 'scientific method', because it suggests that scientific practices come down to one unitary thing, and because it suggests that that thing is fundamentally different from everyday forms of investigation. But your question still stands: is pure mathematics a science, alongside physics, biology, psychology, etc, or is it fundamentally different?

Mathematics is of course used in other sciences, but that is not distinctive. Physics is used in chemistry, and chemistry is used in biology, to give just two other examples. Nor is mathematics clearly different on the grounds that it does not study real objects, since some philosophers and mathematicians believe that mathematical objects and relations are real, though abstract. But many (though by no means all) philosophers would say that mathematics is nevertheless fundamentally different from the (other) sciences, because it is not empirical: we do not need to test its results by means of observation and experiment.

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