It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting the constructive stance in mathematics?
What are the reasons for not supporting constructive mathematics?
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$\begingroup$
logic
soft-question
philosophy
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16Because it requires more work, and mathematicians are by nature lazy? – 2012-07-22
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16Because it’s incompatible with much of the mathematics that I like. – 2012-07-22
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0@Marc van Leeuwen : Some of us are by nature masochists, and those of us who are actually do constructive mathematics. But there are a few of them, not so many, fair enough... =) – 2012-07-22
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7Because proofs by contradiction are always fun! – 2012-07-22
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5I tend to believe that things can exist, even if we are unable to describe them explicitly. – 2012-07-22
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4I wholeheartedly agree with the commented by Marc and Brian, yet there's one further reason important to me: the excluded middle makes a lot of sense in my mind. For me, it is a completely sound and "logical" assumption and I've no problem at all with it. Just as AC, but even easier to grasp. – 2012-07-22
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2Because those truths that we intuit outstrip those that we can prove, and even more so those, truths that we do actually prove. – 2012-07-22
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2Because classical logic doesn't lead to falsehoods. (On which even ituitionists would agree, though they may dispute whether all it leads to is truth). – 2012-07-22
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2@DonAntonio: I don't understand what you mean - the axiom of choice is obviously true! (On the other hand, the well-ordering principle is obviously false, and I personally don't believe Zorn's Lemma. It is silly.) – 2012-07-23
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1Well @user1729, if AC was "obviously true" why then there are *mathematicians* who don't accept it? Perhaps what's "obvious" to you isn't to others...? OTOH, in ZF, the well-ordering principle is equivalent with, oh paradox!, both AC and Zorn's Lemma, so your calling the last one "silly" doesn't make you look specially sharp, does it? Now, if you want to throw away ZF and work under other system go ahead and be anyone's guest. – 2012-07-23
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11@DonAntonio: "The Axiom of Choice is clearly true, the well-ordering principle is clearly false, and nobody really knows about Zorn's lemma" is an old math joke. – 2012-07-23
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0@MichaelGreinecker, I know that, thanks. Yet user 1729 changed in such a way (in particular the adding "it is silly") that made it look, at least in my eyes, as no particularly "jokely". Anyway I shall have my irony-sarcasm glands checked next week. – 2012-07-23
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1@DonAntonio: Deadpan humour is the best kind of humour. I was in a talk last week where Martin Bridson nonchalantly described studying the Nilpotent Genus, I believe it was, of a group as "a disease he caught from Baumslag". Que shock then laughter from us graduate students hiding at the back... – 2012-07-23
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3@Brian M. Scott: most constructive logics have the property that the things they prove are a proper subset of the things provable classically. These logics are not in any way incompatible with usual mathematics, they are just weaker. – 2012-07-23
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1@BrianM.Scott: I think that holds true for much of what i done under the banner of constructivism, especially the things in the tradition of Bishop. But Brouwers intuitionism was very different. – 2012-07-23
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0@Carl: Weak enough to be incompatible with what I want to do. I like non-constructive existence proofs. I insist on the axiom of choice. – 2012-07-23
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1@Brian M. Scott: many constructive systems include the axiom of choice, it's a completely separate issue. – 2012-07-23
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0@Carl: Not in the sense of *constructive* with which I’m familiar and was using the term. (And I certainly wouldn’t apply the term to anything that permitted the use of AC.) – 2012-07-23
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0@Carl, or arguably sometimes stronger since we can interpret classical logic in intuitionistic logic. It is a more expressive language so there are more distinctions and more unprovable equivalences. (I sometimes wonder what would have happened if constructivists used new logical symbols in place of classical ones. :) – 2012-07-24
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4I find the wording of the question a little strange. I have no *stance* myself on any of these issues. I request different kinds of proofs of facts depending on what I want. Sometimes I want a constructive procedure, sometimes I don't care. It really depends on the purpose of the proposition as to which type of proof you want or which kind of formalism you want to live in. – 2012-07-24
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0what is constructive mathematics? – 2013-09-14
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0i think by constructive (and excluded middle) you refer to intuitionism specifically, which accepts excluded middle but not un-critically. For example for very large sets where it is not at all obvious or inferable that this holds.. – 2014-05-27
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0"Supporting" and "adopting" are not synonyms. It would be helpful if you could reconcile the title of your question with the question. I support constructive mathematics, but I don't adopt constructivism as a dogma. – 2016-08-01