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I'm a mathematician working in industry and I'm not at all interested in foundations. That's partly because I trust that all that stuff was worked out in the l

I'm a mathematician working in industry and I'm not at all interested in foundations. That's partly because I trust that all that stuff was worked out in the last century, and partly because there's so much more exciting stuff in math to be interested in instead.

pmoriarty 10y ago
Your belief that foundations were worked out in the last century is completely mistaken. Very serious and concerted efforts at foundations were made then, but they famously and dramatically failed when apparently unreconcilable contradictions were found at the core of these foundations. Then the foundational effort was largely abandoned and mathematicians moved on. I suppose back around that time and up to a certain time after this happened, most mathematicians were aware of this history, but perhaps they aren't any more, if yours is a representative view. So then it comes down to interest. But many mathematicians were interested in foundations around the early part of the 20th Century. Now they're generally not. This shift makes me wonder how much of mathematical foundations are actually "inherently interesting" (if there is such a thing) to mathematicians and how much has to do with fashion, and if it foundational projects among mathematicians will ever become fashionable (or "interesting") again.
- GFK_of_xmaspast 10y ago
  Can you name three unreconcilable contradictions in ZFC?
  - pmoriarty 10y ago
    "The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case, and that in both the systems mentioned there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms." From "On Formally Undecidable Propositions of Principia Mathematica And Related Systems" by Kurt Gödel, 1930 https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propos... http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf
    - GFK_of_xmaspast 10y ago
      I am aware of Godel, thanks, and of course that choice / Zorn's / well ordering (not to mention CH) are independent of ZF (and in particular that's why I mentioned ZFC). I'm also willing to accept the existence of an inaccessible cardinal, which bears on ZFC's consistency. Ultimately my belief is this: there was a very real crisis in foundations at the end of the 19th century and over the next several decades this was fixed as best as it could be. The edges of the foundation are not perfect, the edges of the foundation cannot be perfect, but the edges of the foundation have been pushed back so far that for nearly every working mathematician they're good enough. (and if the algebraic geometers need Grothendieck universes, I'm ok with that)

pmoriarty 10y ago

Your belief that foundations were worked out in the last century is completely mistaken. Very serious and concerted efforts at foundations were made then, but they famously and dramatically failed when apparently unreconcilable contradictions were found at the core of these foundations. Then the foundational effort was largely abandoned and mathematicians moved on. I suppose back around that time and up to a certain time after this happened, most mathematicians were aware of this history, but perhaps they aren't any more, if yours is a representative view. So then it comes down to interest. But many mathematicians were interested in foundations around the early part of the 20th Century. Now they're generally not. This shift makes me wonder how much of mathematical foundations are actually "inherently interesting" (if there is such a thing) to mathematicians and how much has to do with fashion, and if it foundational projects among mathematicians will ever become fashionable (or "interesting") again.

GFK_of_xmaspast 10y ago
Can you name three unreconcilable contradictions in ZFC?
- pmoriarty 10y ago
  "The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case, and that in both the systems mentioned there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms." From "On Formally Undecidable Propositions of Principia Mathematica And Related Systems" by Kurt Gödel, 1930 https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propos... http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf
  - GFK_of_xmaspast 10y ago
    I am aware of Godel, thanks, and of course that choice / Zorn's / well ordering (not to mention CH) are independent of ZF (and in particular that's why I mentioned ZFC). I'm also willing to accept the existence of an inaccessible cardinal, which bears on ZFC's consistency. Ultimately my belief is this: there was a very real crisis in foundations at the end of the 19th century and over the next several decades this was fixed as best as it could be. The edges of the foundation are not perfect, the edges of the foundation cannot be perfect, but the edges of the foundation have been pushed back so far that for nearly every working mathematician they're good enough. (and if the algebraic geometers need Grothendieck universes, I'm ok with that)