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Often in mathematics one constructs a set of some sort, let's name it $A$. We've constructed it in an abstract way, so, a priori, structural aspects of $A$ are yet unknown to us, until we prove them. Lets say that, after some effort, we've proven that $A$ in fact has a set of properties $P$. If we happen to have previously studied, in general, the implications of the properties $P$, then we can now apply everything we studied about $P$ to $A$, and we all live happily ever after. It generally works this way, and people are considered satisfied with applying what we know about $P$ in $A$ if and only if we have proven that we can. Sometimes we can't, but in these cases there's an massive collective effort to do so.

In physics, is this the case? I'm very, very interested in finding out if anyone has made efforts to justify why we use mathematics to study the world outside of our minds. Because in doing so, we are treating the world as if it were $A$ in the example, and we are applying $P$, but without justification. Most likely there isn't such justification, because it'll all bubble down to some parmenides-style paradox about what-is-not, maybe... but I still would like to know about attempts, whether historical or modern.

Any books on the matter? online pdf's? I won't consider it an invalid answer if someone recommends some philosopher or some other, or literarian, or who-be-it.

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There has been extensive discussion, of course. You will find a lot of hits if you search under "the unreasonable effectiveness of mathematics." – André Nicolas Jan 10 '14 at 23:47
@GPerez Just fyi (I haven't read much on this post/comments): The only people who are moderators are the people with diamonds next to their names. They are also found here: – apnorton Jan 11 '14 at 0:51
Serious why so many soft questions lately? – Lost1 Jan 11 '14 at 0:51
@GPerez Putting a hostile sarcastic remark as an edit is usually not the best way to get a question reopened. – Mike Miller Jan 11 '14 at 0:56
Physicists doesn't justify why one use mathematics at all. It is just a tool/language. To physicists, the only gauge is whether it produces a description which matches experiment. No matter how rigorous is a piece of math, if it didn't match experiment, it is not useful. No matter who shabby is a piece of math, if it do match experiment, it will be accepted reluctantly. Mathematicians uses a different gauge, a piece of math is correct or not doesn't depend on anything in physical world. Its correctness depends on and only on pure logic. – achille hui Jan 11 '14 at 1:27
up vote 1 down vote accepted

To the extent I understand your question (I have to protect myself against those waters of accusation too!), it made me think of the following classic:

Mathematics and Plausible Reasoning, by G. Polya. Volume 1: Patterns of Plausible Inference; Volume II: Induction and Analogy in Mathematics.

Polya develops a structure around what he calls undemonstrable inferences. So for example, instead of $A \implies B$ and $\neg B$ together imply $\neg A$, you might have $A \implies B$ and $B$ together imply A more credible.

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There has been considerable discussion in the literature of this problem. One particularly interesting school of thought is the Marburg neo-Kantianism of Hermann Cohen, Ernst Cassirer, and others. From this viewpoint, the division of the problem into $P$ somehow assumed to exist "out there" and $A$ which is a mathematical model thereof is part of the problem, or more precisely "outdated dualistic metaphysics bound to lead into unsolvable, self-inflicted pseudo-problems", as Cassirer put it in 1910. To respond to your request for a reference, I could suggest this recent article.

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