Is there any physical or computational justification for non-constructive axioms such as AC or excluded...












3












$begingroup$


I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.



But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.



I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game.



Essentially, my question boils down to the following:




  1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).


  2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?











share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    Nothing is sufficient to describe reality.
    $endgroup$
    – darij grinberg
    3 hours ago










  • $begingroup$
    Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
    $endgroup$
    – ಠ_ಠ
    3 hours ago






  • 1




    $begingroup$
    It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
    $endgroup$
    – Somos
    2 hours ago






  • 1




    $begingroup$
    I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
    $endgroup$
    – Alexis Hazell
    1 hour ago








  • 2




    $begingroup$
    @AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
    $endgroup$
    – Matt F.
    1 hour ago
















3












$begingroup$


I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.



But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.



I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game.



Essentially, my question boils down to the following:




  1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).


  2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?











share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    Nothing is sufficient to describe reality.
    $endgroup$
    – darij grinberg
    3 hours ago










  • $begingroup$
    Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
    $endgroup$
    – ಠ_ಠ
    3 hours ago






  • 1




    $begingroup$
    It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
    $endgroup$
    – Somos
    2 hours ago






  • 1




    $begingroup$
    I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
    $endgroup$
    – Alexis Hazell
    1 hour ago








  • 2




    $begingroup$
    @AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
    $endgroup$
    – Matt F.
    1 hour ago














3












3








3





$begingroup$


I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.



But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.



I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game.



Essentially, my question boils down to the following:




  1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).


  2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?











share|cite|improve this question











$endgroup$




I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.



But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.



I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game.



Essentially, my question boils down to the following:




  1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).


  2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?








lo.logic soft-question mathematical-philosophy constructive-mathematics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago







ಠ_ಠ

















asked 3 hours ago









ಠ_ಠಠ_ಠ

2,60821535




2,60821535








  • 3




    $begingroup$
    Nothing is sufficient to describe reality.
    $endgroup$
    – darij grinberg
    3 hours ago










  • $begingroup$
    Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
    $endgroup$
    – ಠ_ಠ
    3 hours ago






  • 1




    $begingroup$
    It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
    $endgroup$
    – Somos
    2 hours ago






  • 1




    $begingroup$
    I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
    $endgroup$
    – Alexis Hazell
    1 hour ago








  • 2




    $begingroup$
    @AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
    $endgroup$
    – Matt F.
    1 hour ago














  • 3




    $begingroup$
    Nothing is sufficient to describe reality.
    $endgroup$
    – darij grinberg
    3 hours ago










  • $begingroup$
    Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
    $endgroup$
    – ಠ_ಠ
    3 hours ago






  • 1




    $begingroup$
    It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
    $endgroup$
    – Somos
    2 hours ago






  • 1




    $begingroup$
    I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
    $endgroup$
    – Alexis Hazell
    1 hour ago








  • 2




    $begingroup$
    @AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
    $endgroup$
    – Matt F.
    1 hour ago








3




3




$begingroup$
Nothing is sufficient to describe reality.
$endgroup$
– darij grinberg
3 hours ago




$begingroup$
Nothing is sufficient to describe reality.
$endgroup$
– darij grinberg
3 hours ago












$begingroup$
Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
$endgroup$
– ಠ_ಠ
3 hours ago




$begingroup$
Well obviously physics is a work-in-progress. My question is really if we have any physical or computational justification for some form of choice axiom or excluded middle.
$endgroup$
– ಠ_ಠ
3 hours ago




1




1




$begingroup$
It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
$endgroup$
– Somos
2 hours ago




$begingroup$
It is a mistake to confuse "mathematics" with "reality" in some meaning of the words. Only by clarifying the meaning can some connections and/or comparisons become meaningful. Please tell us what you really mean by "mathematics" and "reality".
$endgroup$
– Somos
2 hours ago




1




1




$begingroup$
I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
$endgroup$
– Alexis Hazell
1 hour ago






$begingroup$
I'd also be interested to know whether there are constructive versions of the fixed-point theorems used in game theory, as referred to by this comment in that thread. (Related: this MO discussion about whether Lawvere's fixed-point theorem implies Brouwer's, which appears to be an open question.)
$endgroup$
– Alexis Hazell
1 hour ago






2




2




$begingroup$
@AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
$endgroup$
– Matt F.
1 hour ago




$begingroup$
@AlexisHazell, yes, you can prove appropriate versions of those fixed-point theorems using Sperner’s Lemma, eg here: hokuriku.me/kiishimizu/open/AAMA.pdf. But as I suggest in my answer, that makes the proofs much messier.
$endgroup$
– Matt F.
1 hour ago










1 Answer
1






active

oldest

votes


















3












$begingroup$

It depends on what you want out of your science!



If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.



If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.



But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.



Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.



So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "504"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323616%2fis-there-any-physical-or-computational-justification-for-non-constructive-axioms%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    It depends on what you want out of your science!



    If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.



    If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.



    But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.



    Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.



    So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      It depends on what you want out of your science!



      If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.



      If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.



      But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.



      Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.



      So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        It depends on what you want out of your science!



        If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.



        If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.



        But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.



        Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.



        So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.






        share|cite|improve this answer









        $endgroup$



        It depends on what you want out of your science!



        If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.



        If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.



        But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.



        Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.



        So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 1 hour ago









        Matt F.Matt F.

        6,60311743




        6,60311743






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to MathOverflow!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323616%2fis-there-any-physical-or-computational-justification-for-non-constructive-axioms%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Knooppunt Holsloot

            Altaar (religie)

            Gregoriusmis