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Because there are many philosophical studies at present, like

  • philosophical study of education,

  • philosophical study of judgement, etc.

Therefore, please explain to me what the philosophical study of something means.

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    Philosophy = reflection on the meaning of some basic concept, like Good, Justice, Knowledge, etc. Also the study of fundamental principles of some basic human activity : science, art, politics, etc. – Mauro ALLEGRANZA Apr 13 '19 at 12:49
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Philosophical study means study of a topic at the most fundamental level. We explore all core aspects of the topic in detail so as to fully understand it. If there are more than one widely accepted philosophical theories pertaining to a subject, they exist side by side because both lead to same conclusions which are consistent with what we believe, expect or experience.

In this sense philosophy is more like widely agreed opinions -backed by reasoning or empirical evidence. For example, one may say mathematics is Platonic (that it exists independently of us) or mathematics is a construction of human mind. Here we are exploring the nature of existence of Mathematics. In philosophy we explore ideas like: Why does this exist? Why does this work? What does it really mean when we say something? Why should we believe in a mathematical proof? By simply reading this paragraph, how and why are you able to able to understand what I mean?

More importantly, philosophical viewpoints can often form a (metaphorical) base for further work of philosophers, mathematicians, logicians, etc. For example, Alan Turing was a proponent of Formalism, while Ludwig Wittgenstein believed in human/social construction of mathematics/language. Accordingly, their works are milestones in their respective philosophies -Turing Machines and Wittgenstein's Philosophical Investigations.

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A philosophical study is the analysis of the fundamental nature of one or more aspects of the world in which we live... Any and every aspect of the world is subject to scrutiny for the purposes of gaining insight and deepening the knowledge of ourselves and the world around us.

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