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You make a couple of basic mistakes in your question. Perhaps you should correct them and ask again because I am not entirely sure what it is you are asking:

1. Topos theory does not "freely use $P \lor \lnot P$", and neither does synthetic differential geometry. In fact, topos theorists are quite careful about not using the law of excluded middle, while synthetic differential geometry proves the negation of the law of excluded middle.

2. As far as I know, the law of excluded middle is $P \lor \lnot P$, while the law of non-contradiction is $\lnot (P \land \lnot P)$. These two are not equivalent (unless you already believe in the law of excluded middle, in which case the whole discussion is trivial). The principle of non-contradiction is of course intuitionistically valid. So you seem to be confusing two different logical principles.

If I had to guess what you asked, I would say you are wondering why anyone in their right mind would want to be agnostic about the law of excluded middle (intuitionistic logic) or even deny it (synthetic differential geometry). Aren't people who do so just plain crazy?

To understand why someone might work without the law of excluded middle, the best thing is to study their theories. Probably you cannot afford to devote several years of your life to the study of topos theory. For an executive summary of synthetic differential geometry and its interplay with logic I recommend John Bell's texts on synthetic differential geometry, such as this one.

Let me try an analogy. Imagine a mathematician who studies commutative groups and has never heard of the non-commutative ones. One day he meets another mathematician who shows him non-commutative groups. How will the first mathematician react? I imagine he will go through all the usual phases:

1. Denial: these are not groups!
2. Anger: why are you destroying my groups? I hate you!
3. Bargaining: can we at least analyze non-commutative group in terms of their "commutative representations" (whatever that would mean)?
4. Depression: this is hopeless, I wasted my life studying the wrong groups. I might as well study point-set topology.
5. Acceptance: it's kind of cool that the symmetries of a cube form a group.

I am at stage 5 with regards to intuitionistic logic. Where are you?

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You make a couple of basic mistakes in your question. Perhaps you should correct them and ask again because I am not entirely sure what it is you are asking:

1. Topos theory does not "freely use $P \lor \lnot P$", and neither does synthetic differential geometry. In fact, topos theorists are quite careful about not using the law of excluded middle, while synthetic differential geometry proves the negation of the law of excluded middle.

2. As far as I know, the law of excluded middle is $P \lor \lnot P$, while the law of non-contradiction is $\lnot (P \land \lnot P)$. These two are not equivalent (unless you already believe in the law of excluded middle, in which case the whole discussion is trivial). The principle of non-contradiction is of course intuitionistically valid. So you seem to be confusing two different logical principles.

If I had to guess what you asked, I would say you are wondering why anyone in their right mind would want to be agnostic about the law of excluded middle (intuitionistic logic) or even deny it (synthetic differential geometry). Aren't people who do so just plain crazy?

To understand why someone might work without the law of excluded middle, the best thing is to study their theories. Probably you cannot afford to devote several years of your life to the study of topos theory. For an executive summary of synthetic differential geometry and its interplay with logic I recommend John Bell's texts on synthetic differential geometry, such as this one.

Let me try an analogy. Imagine a mathematician who studies commutative groups and has never heard of the non-commutative ones. One day he meets another mathematician who shows him non-commutative groups. How will the first mathematician react? I imagine he will go through all the usual phases:

1. Denial: these are not groups!
2. Anger: why are you destroying my groups? I hate you!
3. Bargaining: can we at least analyze non-commutative group in terms of their "commutative representations" (whatever that would mean)?
4. Depression: this is hopeless, I wasted my life studying the wrong groups. I might as well study point-set topology.
5. Acceptance: it's kind of cool that the symmetries of a cube form a group.

I am at stage 5 with regards to intuitionistic logic. Where are you?