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On the analogy of castles, I asked a professor whether Godel’s Incompleteness theorems affects their work. She told me Mathematics is this great castle, where i

On the analogy of castles, I asked a professor whether Godel’s Incompleteness theorems affects their work. She told me Mathematics is this great castle, where in one quadrant you have the number theorist, in another the topologists. Every decade or century, the logicians would come running up from the basement with results confronting the foundations of math, but everyone else continued with their routine

pasabagi 7y ago
Perhaps the point is that the basement of a castle is also called a dungeon.
bitL 7y ago
Yeah, swapping out currently dominant logic with another one would redo the whole castle, like when a castle built in Gothic architecture is rebuilt into Baroque with some parts left intact. However, mainstream is so dominant and "better logics" either don't have sufficient manpower, are more complicated, are weaker or impractical, so that they never develop unless they solve something extraordinary quickly.
pmoriarty 7y ago
My memory from the logic classes I've taken was that after Godel, mathematicians largely abandoned Hilbert's, Russell's and Whitehead's ambition of founding mathematics on logic, and largely grew disinterested in foundational issues as a whole. As such, mathematics doesn't have a solid foundation, and most contemporary mathematicians seem to be ok with that, as long as it works and they can get interesting results.
- dwohnitmok 7y ago
  I agree with the rest of what you wrote, but doesn't have a solid foundation seems a bit far. Mathematicians hand-wave that their proofs can be reduced to ZFC set theory and despite reasonable wariness about that hand-waving, it has so far been borne out by all efforts at deeper inspection all the way up to full machine-checked formalization.
  - pmoriarty 7y ago
    Although I am starting to get out of my depth, my impression was that ZFC itself is (per Godel's results) inconsistent or incomplete. Here is some research to this effect: If this is indeed the case (as it seems to my unprofessional first glance to be) then mathematics does not in fact have a solid foundation, despite its appeal to ZFC. I'd love to hear a professional logician weigh in on this, if there are any on here. [1] - https://www.researchgate.net/profile/Costas_Drossos/post/Is_the_set_theory_ZFC_consistent2/attachment/59d62a61c49f478072e9cefe/AS:272539979321361@1441990002786/download/Consistency+and+Incompleteness+of_ZFC.pdf https://www.researchgate.net/profile/Costas_Drossos/post/Is_... [2] - https://arxiv.org/abs/1203.0494 https://arxiv.org/abs/1203.0494
    - Jweb_Guru 7y ago
      The short answer is "incompleteness is not a big deal" and the slightly longer answer is "if it's inconsistent, the proof of inconsistency is devilishly hard to find." If it does exist, the inconsistent part seems likely to be sufficiently esoteric that it could be "built around."
    - dwohnitmok 7y ago
      ZFC is incomplete. There are many celebrated results that are independent of ZFC (continuum hypothesis, large cardinals (a lot aren't even known to be consistent with ZFC), Suslin's Hypothesis, etc. However, by Godel we know this is true for any reasonable foundation of mathematics (i.e. anything capable of expressing the usual version of arithmetic). So it is not something that particularly troubles either logicians or other mathematicians. Godel's results also show that we cannot use ZFC to prove that ZFC is consistent. However, it is strongly suspected that ZFC is consistent. In practice, very few of the results independent of ZFC affect day to day mathematical work unless you're a logician or set theorist (which honestly start to blend into each other).
      - dwohnitmok 7y ago
        Oh I forgot to add. The inability to prove its own consistency is also something any reasonable foundation will have.
dwohnitmok 7y ago
AFAICT mathematical logic has always felt like a curious side show in the greater math community. If you were to ask professional mathematicians to list off all the axioms of ZFC they'd probably shrug if they didn't get all of them. They probably wouldn't know about the more exotic parts of set theory such as large cardinals nor would they would probably know too much about alternate foundations of mathematics. They might not be familiar at all with the specifics of model theory. The logical foundations of mathematics have always had a much smaller impact on the day to day lives of mathematicians than their foundational nature might suggest.
- whatshisface 7y ago
  I wonder what fraction of web programmers understand transistor physics?
  - dwohnitmok 7y ago
    Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use (nor honestly do they need to for the most part). Nonetheless, it's perhaps surprising for a beginning student of pure mathematics when all they can see is the sky-high demand for rigor that "professional" pure mathematics is seemingly content to hand-wave its foundations. (For the most part this hand-waving is the postrigorous stage described by Terence Tao https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-... rather than the pre-rigorous hand-waving of the student).
  - segfaultbuserr 7y ago
    > what fraction of web programmers understand transistor physics? I'm not an engineer, but I imagine that even most professional electrical/electronic engineers do not understand transistor physics in full details. In engineering, a transistor is essentially treated as a blackbox, and its behavior is described and approximated by various small-signal and large-signal models (i.e. treated as a lumped component), with their parameters characterized empirically by vendors through experiments. How exactly things work at atomic or quantum level is essentially for a transistor to work, yet, a subject of study unrelated to electrical engineering. On the other hand, I imagine there is no shortage of EEs who have studied a physics major, or EEs with a background of semiconductor physics - they can understand transistor physics really well. So we can say the relation between EE and transistor physics sounds a lot similar to mathematicians and logicians, it's a good analogy.
perl4ever 7y ago
My impression is that today we know that there can be no single foundation, which means that there is no way to undermine all areas at once.

pasabagi 7y ago

Perhaps the point is that the basement of a castle is also called a dungeon.

bitL 7y ago

Yeah, swapping out currently dominant logic with another one would redo the whole castle, like when a castle built in Gothic architecture is rebuilt into Baroque with some parts left intact. However, mainstream is so dominant and "better logics" either don't have sufficient manpower, are more complicated, are weaker or impractical, so that they never develop unless they solve something extraordinary quickly.

pmoriarty 7y ago

My memory from the logic classes I've taken was that after Godel, mathematicians largely abandoned Hilbert's, Russell's and Whitehead's ambition of founding mathematics on logic, and largely grew disinterested in foundational issues as a whole. As such, mathematics doesn't have a solid foundation, and most contemporary mathematicians seem to be ok with that, as long as it works and they can get interesting results.

dwohnitmok 7y ago
I agree with the rest of what you wrote, but doesn't have a solid foundation seems a bit far. Mathematicians hand-wave that their proofs can be reduced to ZFC set theory and despite reasonable wariness about that hand-waving, it has so far been borne out by all efforts at deeper inspection all the way up to full machine-checked formalization.
- pmoriarty 7y ago
  Although I am starting to get out of my depth, my impression was that ZFC itself is (per Godel's results) inconsistent or incomplete. Here is some research to this effect: If this is indeed the case (as it seems to my unprofessional first glance to be) then mathematics does not in fact have a solid foundation, despite its appeal to ZFC. I'd love to hear a professional logician weigh in on this, if there are any on here. [1] - https://www.researchgate.net/profile/Costas_Drossos/post/Is_the_set_theory_ZFC_consistent2/attachment/59d62a61c49f478072e9cefe/AS:272539979321361@1441990002786/download/Consistency+and+Incompleteness+of_ZFC.pdf https://www.researchgate.net/profile/Costas_Drossos/post/Is_... [2] - https://arxiv.org/abs/1203.0494 https://arxiv.org/abs/1203.0494
  - Jweb_Guru 7y ago
    The short answer is "incompleteness is not a big deal" and the slightly longer answer is "if it's inconsistent, the proof of inconsistency is devilishly hard to find." If it does exist, the inconsistent part seems likely to be sufficiently esoteric that it could be "built around."
  - dwohnitmok 7y ago
    ZFC is incomplete. There are many celebrated results that are independent of ZFC (continuum hypothesis, large cardinals (a lot aren't even known to be consistent with ZFC), Suslin's Hypothesis, etc. However, by Godel we know this is true for any reasonable foundation of mathematics (i.e. anything capable of expressing the usual version of arithmetic). So it is not something that particularly troubles either logicians or other mathematicians. Godel's results also show that we cannot use ZFC to prove that ZFC is consistent. However, it is strongly suspected that ZFC is consistent. In practice, very few of the results independent of ZFC affect day to day mathematical work unless you're a logician or set theorist (which honestly start to blend into each other).
    - dwohnitmok 7y ago
      Oh I forgot to add. The inability to prove its own consistency is also something any reasonable foundation will have.

dwohnitmok 7y ago

AFAICT mathematical logic has always felt like a curious side show in the greater math community. If you were to ask professional mathematicians to list off all the axioms of ZFC they'd probably shrug if they didn't get all of them. They probably wouldn't know about the more exotic parts of set theory such as large cardinals nor would they would probably know too much about alternate foundations of mathematics. They might not be familiar at all with the specifics of model theory. The logical foundations of mathematics have always had a much smaller impact on the day to day lives of mathematicians than their foundational nature might suggest.

whatshisface 7y ago
I wonder what fraction of web programmers understand transistor physics?
- dwohnitmok 7y ago
  Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use (nor honestly do they need to for the most part). Nonetheless, it's perhaps surprising for a beginning student of pure mathematics when all they can see is the sky-high demand for rigor that "professional" pure mathematics is seemingly content to hand-wave its foundations. (For the most part this hand-waving is the postrigorous stage described by Terence Tao https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-... rather than the pre-rigorous hand-waving of the student).
- segfaultbuserr 7y ago
  > what fraction of web programmers understand transistor physics? I'm not an engineer, but I imagine that even most professional electrical/electronic engineers do not understand transistor physics in full details. In engineering, a transistor is essentially treated as a blackbox, and its behavior is described and approximated by various small-signal and large-signal models (i.e. treated as a lumped component), with their parameters characterized empirically by vendors through experiments. How exactly things work at atomic or quantum level is essentially for a transistor to work, yet, a subject of study unrelated to electrical engineering. On the other hand, I imagine there is no shortage of EEs who have studied a physics major, or EEs with a background of semiconductor physics - they can understand transistor physics really well. So we can say the relation between EE and transistor physics sounds a lot similar to mathematicians and logicians, it's a good analogy.

perl4ever 7y ago

My impression is that today we know that there can be no single foundation, which means that there is no way to undermine all areas at once.