4 ms·

The other day I realized that a man-made law is also a bit like mathematics or computer software. It is carefully designed and constructed. Ideally, it is inten

by euske 11y ago

The other day I realized that a man-made law is also a bit like mathematics or computer software. It is carefully designed and constructed. Ideally, it is intended to work like a machine with as little room for human discretion as possible. And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences.

deleted 11y ago
[deleted]
johncolanduoni 11y ago
> And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences. Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. All of these fields may add definitions, but these are just shorthand; the are conservative and have no actual consequences. It is more like fixing Newton's laws and then experimenting with all the machines you can build with them. [1]: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
- xamuel 11y ago
  Most mathematicians don't spend their time considering extensions of ZFC, but mathematicians are always adding axioms in a different sense. Add a commutativity axiom to group theory and you get commutative group theory. Add an axiom that "the space has total measure 1" to measure theory and you get probability theory. And so on.
  - johncolanduoni 11y ago
    These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. They are simply part of the definition of a group, commutative group, etc. Since they are just part of the definition of the structure, choosing the "wrong axioms" will only result in a structure which has no actual instances, not in inconsistency of the underlying theory. As such, they most certainly don't have "far reaching consequences", except that some axioms may produce uninteresting structures.
    - ihm 11y ago
      > These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct". For example, one could say that general group theory is studying the theory (in the logic sense) of a group (usually the second order theory rather than first order, but you get idea). Then adding assumptions about the structure of the group under consideration is like adding axioms and looking at the new theory.
    - deleted 11y ago
      [deleted]
    - nmrm2 11y ago
      This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. And, doing some obscure work on an inconsistent foundations that no body uses is exactly as harmless as doing some obscure work on a structure without elements.
      - johncolanduoni 11y ago
        > This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. They may not do so explicitly, but it is easy to see where the axioms become important when dealing with mathematical structures. For example, if you are considering the group of integers with addition, the existence of this group is predicated on the existence of the integers, which is predicated on the existence of the natural numbers as a set, which is an axiom of ZFC (axiom of infinity[1]). The case for other foundations is similar. > But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. No, because when studying a mathematical structure we always consider concrete instances of it, which are usually in fact the motivation for the definition of the structure. So mathematics based on possibly (relatively) inconsistent structures (other than foundational systems) is not really a thing; group theory was not initiated prior to the discovery of structures satisfying the groups axioms, and neither was the study of rings, fields, modules, topological spaces etc. On the other hand, we cannot consider models of ZFC unless we build it within the assumptions of a foundational system (possibly ZFC itself). So at some point we have to pick a set of axioms to be the bottom of our ladder of turtles, and the axioms of that system are very special indeed. Another difference is that the axioms of a group (or a field, or a topological space) cannot stand on their own; they clearly reference sets and functions. ZFC's axioms only reference sets, all of the fundamental properties of which are enumerated in the axioms themselves. If you gave an (english speaking) alien the definition of a group, he would need to ask you what the definition of sets and functions were. So even if we wanted the group axioms to be our foundations, this would not be possible. [1] : https://en.wikipedia.org/wiki/Axiom_of_infinity https://en.wikipedia.org/wiki/Axiom_of_infinity
- eli_gottlieb 11y ago
  >Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. Except for all the non-ZFC foundations, like type theory or category-theoretic foundations.
  - cwzwarich 11y ago
    Relatively speaking, there are very few mathematicians using those systems to discover new mathematics as opposed to studying them for their foundational interest.
  - johncolanduoni 11y ago
    They (and those who study ZFC extensions) are the reason I said "virtually"; they are most certainly a small minority among mathematicians. I myself have been studying homotopy type theory as of late.
Someone 11y ago
Changes in law (almost) never add axioms because that would have global effects; instead, they tend to add folds to the manifold that law is. And many folds there certainly are. It is hard to find anything in law that holds universally. Examples (I’m picking mostly US law here, but I’m sure similar exceptions exist elsewhere): - everybody can vote? Well, we have https://en.wikipedia.org/wiki/Felony_disenfranchisement https://en.wikipedia.org/wiki/Felony_disenfranchisement and, on the other end of the spectrum, in the UK "Although the law relating to elections does not specifically prohibit the Sovereign from voting in a general election or local election, it is considered unconstitutional for the Sovereign and his or her heir to do so” (http://www.royal.gov.uk/MonarchUK/QueenandGovernment/Queenandvoting.aspx http://www.royal.gov.uk/MonarchUK/QueenandGovernment/Queenan...) - everybody with a sufficiently high income must pay social security taxes? Not if you’re member of certain religious groups (https://faq.ssa.gov/link/portal/34011/34019/Article/3821/Are-members-of-religious-groups-exempt-from-paying-Social-Security-taxes https://faq.ssa.gov/link/portal/34011/34019/Article/3821/Are...) Even the universal declaration of human rights (http://www.un.org/en/documents/udhr/ http://www.un.org/en/documents/udhr/) often has small exceptions. For example: - "Everyone has the right to take part in the government of his country, directly or through freely chosen representatives"? Not quite in the USA, as one must be born in the USA to become president. - "higher education shall be equally accessible to all on the basis of merit": questionable in many countries, given the costs. It seems there’s no rule so universal that it doesn’t have some exception. That, IMO, makes law so different from math that any analogy is useless.
- nmrm2 11y ago
  The analogy is useless, but not for this reason. If it were just that the axioms of law are more complicated in the sense of being more conditional, then law would be extremely math-like. After all, theorems with lots of conditional properties (e.g., very common in certain types of algebra, a lot of dynamics, any programming language theory, etc.) are still mathematical theorems. Law is distinguished from Math in two ways: 1) Law has more politics and isn't immune from politics. Mathematical truth, at least, is. 2) In law, the atomic propositions are always difficult to quantify and require human judgement; there is rarely an algorithm for weighing the evidence and determining guilt (or constitutionality, etc).
superuser2 11y ago
>Ideally, it is intended to work like a machine with as little room for human discretion as possible Actually, these laws are the most dystopian. One concept that seems to have this design goal is strict liability: what you intended, what you believed, what elaborate conspiracies were created to deceive you, none of that matters. If you did it, you are guilty and will be punished, full stop. This is attractive because it's much harder to prove beyond a reasonable doubt that someone did something "with malice" or "intentionally" than to prove that they did it. This tends to show up around youth sexuality, probably because that's an area where social norms have changed quite a bit. It doesn't matter if the 17.5-year-old you slept with presented a fake ID saying she was 18, took the lead in all sexual activity, etc. It doesn't matter if she shows up in court and begs the judge to let you go. People whose 18th birthday has not elapsed cannot give consent, full stop. Sex without consent is rape, rapists are bad people and should be made to suffer, so you're going to prison and then on to the sex offender registry. Your judge, and the judges you appeal to, all think this is ridiculous, but their hands are tied because the law was written like a computer program that failed to consider edge cases. (Literally edge cases - those that fall near numerical boundaries. Some states have patched their legal systems to allow consent between partners who are close in age but under the normal age of consent. Some have, but only for heterosexual couples. Some still haven't. My state actually tried and convicted two teenagers as adults for raping each other at the same time.) Same with child porn. Child porn laws were created a time when taking and distributing a picture could only be part of a commercial publishing operation. Anyone who creates or possesses an image of a nude child is guilty of a child pornography offense. Seems reasonable. Now every teenager has an internet-connected camera in their process, and we found an edge case we forgot: the child pornographer is the child him/herself and the consumer is her long-term boyfriend, also underage. Still just as illegal and the same punishment is required. You would hope prosecutors would turn a blind eye, but "the law is the law." These rather salacious examples are the most high-profile, but I'm sure the same problems happens in more mundane areas of law as well. At some point you have to trust judges and you have to empower bureaucrats to help people get out of obviously ridiculous situations. Laws are much more difficult to change than computer programs; there needs to be a reasonable amount of discretion for a manual override until the law can be fixed.

by euske 11y ago

deleted 11y ago

[deleted]

johncolanduoni 11y ago

> And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences. Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. All of these fields may add definitions, but these are just shorthand; the are conservative and have no actual consequences. It is more like fixing Newton's laws and then experimenting with all the machines you can build with them. [1]: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

xamuel 11y ago
Most mathematicians don't spend their time considering extensions of ZFC, but mathematicians are always adding axioms in a different sense. Add a commutativity axiom to group theory and you get commutative group theory. Add an axiom that "the space has total measure 1" to measure theory and you get probability theory. And so on.
- johncolanduoni 11y ago
  These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. They are simply part of the definition of a group, commutative group, etc. Since they are just part of the definition of the structure, choosing the "wrong axioms" will only result in a structure which has no actual instances, not in inconsistency of the underlying theory. As such, they most certainly don't have "far reaching consequences", except that some axioms may produce uninteresting structures.
  - ihm 11y ago
    > These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct". For example, one could say that general group theory is studying the theory (in the logic sense) of a group (usually the second order theory rather than first order, but you get idea). Then adding assumptions about the structure of the group under consideration is like adding axioms and looking at the new theory.
  - deleted 11y ago
    [deleted]
  - nmrm2 11y ago
    This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. And, doing some obscure work on an inconsistent foundations that no body uses is exactly as harmless as doing some obscure work on a structure without elements.
    - johncolanduoni 11y ago
      > This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. They may not do so explicitly, but it is easy to see where the axioms become important when dealing with mathematical structures. For example, if you are considering the group of integers with addition, the existence of this group is predicated on the existence of the integers, which is predicated on the existence of the natural numbers as a set, which is an axiom of ZFC (axiom of infinity[1]). The case for other foundations is similar. > But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. No, because when studying a mathematical structure we always consider concrete instances of it, which are usually in fact the motivation for the definition of the structure. So mathematics based on possibly (relatively) inconsistent structures (other than foundational systems) is not really a thing; group theory was not initiated prior to the discovery of structures satisfying the groups axioms, and neither was the study of rings, fields, modules, topological spaces etc. On the other hand, we cannot consider models of ZFC unless we build it within the assumptions of a foundational system (possibly ZFC itself). So at some point we have to pick a set of axioms to be the bottom of our ladder of turtles, and the axioms of that system are very special indeed. Another difference is that the axioms of a group (or a field, or a topological space) cannot stand on their own; they clearly reference sets and functions. ZFC's axioms only reference sets, all of the fundamental properties of which are enumerated in the axioms themselves. If you gave an (english speaking) alien the definition of a group, he would need to ask you what the definition of sets and functions were. So even if we wanted the group axioms to be our foundations, this would not be possible. [1] : https://en.wikipedia.org/wiki/Axiom_of_infinity https://en.wikipedia.org/wiki/Axiom_of_infinity
eli_gottlieb 11y ago
>Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. Except for all the non-ZFC foundations, like type theory or category-theoretic foundations.
- cwzwarich 11y ago
  Relatively speaking, there are very few mathematicians using those systems to discover new mathematics as opposed to studying them for their foundational interest.
- johncolanduoni 11y ago
  They (and those who study ZFC extensions) are the reason I said "virtually"; they are most certainly a small minority among mathematicians. I myself have been studying homotopy type theory as of late.

Someone 11y ago

Changes in law (almost) never add axioms because that would have global effects; instead, they tend to add folds to the manifold that law is. And many folds there certainly are. It is hard to find anything in law that holds universally. Examples (I’m picking mostly US law here, but I’m sure similar exceptions exist elsewhere): - everybody can vote? Well, we have https://en.wikipedia.org/wiki/Felony_disenfranchisement https://en.wikipedia.org/wiki/Felony_disenfranchisement and, on the other end of the spectrum, in the UK "Although the law relating to elections does not specifically prohibit the Sovereign from voting in a general election or local election, it is considered unconstitutional for the Sovereign and his or her heir to do so” (http://www.royal.gov.uk/MonarchUK/QueenandGovernment/Queenandvoting.aspx http://www.royal.gov.uk/MonarchUK/QueenandGovernment/Queenan...) - everybody with a sufficiently high income must pay social security taxes? Not if you’re member of certain religious groups (https://faq.ssa.gov/link/portal/34011/34019/Article/3821/Are-members-of-religious-groups-exempt-from-paying-Social-Security-taxes https://faq.ssa.gov/link/portal/34011/34019/Article/3821/Are...) Even the universal declaration of human rights (http://www.un.org/en/documents/udhr/ http://www.un.org/en/documents/udhr/) often has small exceptions. For example: - "Everyone has the right to take part in the government of his country, directly or through freely chosen representatives"? Not quite in the USA, as one must be born in the USA to become president. - "higher education shall be equally accessible to all on the basis of merit": questionable in many countries, given the costs. It seems there’s no rule so universal that it doesn’t have some exception. That, IMO, makes law so different from math that any analogy is useless.

nmrm2 11y ago
The analogy is useless, but not for this reason. If it were just that the axioms of law are more complicated in the sense of being more conditional, then law would be extremely math-like. After all, theorems with lots of conditional properties (e.g., very common in certain types of algebra, a lot of dynamics, any programming language theory, etc.) are still mathematical theorems. Law is distinguished from Math in two ways: 1) Law has more politics and isn't immune from politics. Mathematical truth, at least, is. 2) In law, the atomic propositions are always difficult to quantify and require human judgement; there is rarely an algorithm for weighing the evidence and determining guilt (or constitutionality, etc).

superuser2 11y ago

>Ideally, it is intended to work like a machine with as little room for human discretion as possible Actually, these laws are the most dystopian. One concept that seems to have this design goal is strict liability: what you intended, what you believed, what elaborate conspiracies were created to deceive you, none of that matters. If you did it, you are guilty and will be punished, full stop. This is attractive because it's much harder to prove beyond a reasonable doubt that someone did something "with malice" or "intentionally" than to prove that they did it. This tends to show up around youth sexuality, probably because that's an area where social norms have changed quite a bit. It doesn't matter if the 17.5-year-old you slept with presented a fake ID saying she was 18, took the lead in all sexual activity, etc. It doesn't matter if she shows up in court and begs the judge to let you go. People whose 18th birthday has not elapsed cannot give consent, full stop. Sex without consent is rape, rapists are bad people and should be made to suffer, so you're going to prison and then on to the sex offender registry. Your judge, and the judges you appeal to, all think this is ridiculous, but their hands are tied because the law was written like a computer program that failed to consider edge cases. (Literally edge cases - those that fall near numerical boundaries. Some states have patched their legal systems to allow consent between partners who are close in age but under the normal age of consent. Some have, but only for heterosexual couples. Some still haven't. My state actually tried and convicted two teenagers as adults for raping each other at the same time.) Same with child porn. Child porn laws were created a time when taking and distributing a picture could only be part of a commercial publishing operation. Anyone who creates or possesses an image of a nude child is guilty of a child pornography offense. Seems reasonable. Now every teenager has an internet-connected camera in their process, and we found an edge case we forgot: the child pornographer is the child him/herself and the consumer is her long-term boyfriend, also underage. Still just as illegal and the same punishment is required. You would hope prosecutors would turn a blind eye, but "the law is the law." These rather salacious examples are the most high-profile, but I'm sure the same problems happens in more mundane areas of law as well. At some point you have to trust judges and you have to empower bureaucrats to help people get out of obviously ridiculous situations. Laws are much more difficult to change than computer programs; there needs to be a reasonable amount of discretion for a manual override until the law can be fixed.