8 ms·

> In a similar way, mathematics is like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate

> In a similar way, mathematics is like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate truth. There are a couple foundations for mathematics. The traditional one is set theory[1], and there are some newfangled ones based on type theory[2]. [1] https://en.wikipedia.org/wiki/Set_theory https://en.wikipedia.org/wiki/Set_theory "Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice" [2] https://en.wikipedia.org/wiki/Type_theory https://en.wikipedia.org/wiki/Type_theory "a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics"

sewercake 8y ago
I'm speaking out of my expertise here, but I think he's referring to Godel's incompleteness theorum/the halting problem, when he says 'no basis'.
- bloaf 8y ago
  It could also be the admission that you can't beat pure skepticism. That is, you can't actually beat Descartes' deceiving demon; you need to start any line of reasoning with an assumption that your reasoning is somewhat sound. When you're starting out to build a foundation, you have no basis for making that assumption.
  - justinpombrio 8y ago
    We're up to three completely different interpretations of "like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate truth". My takeaway: be wary of analogies; they will be misunderstood.
  - myWindoonn 8y ago
    You grokked it. Imagine standing on a vast desert. You have, floating above you, the HYPERCUBE. You could choose to climb up into the HYPERCUBE, but it has no exits. In a serious, non-trivial sense, mathematics is the pattern science. If you choose to believe that reality has underlying patterns, then mathematics is unavoidable. To respond to the grandparent, we can put sets on categories, types on categories, categories on sets, logic on sets, logic on categories, etc. Who cares which foundations you pick, when they all can build from each other?

sewercake 8y ago

I'm speaking out of my expertise here, but I think he's referring to Godel's incompleteness theorum/the halting problem, when he says 'no basis'.

bloaf 8y ago
It could also be the admission that you can't beat pure skepticism. That is, you can't actually beat Descartes' deceiving demon; you need to start any line of reasoning with an assumption that your reasoning is somewhat sound. When you're starting out to build a foundation, you have no basis for making that assumption.
- justinpombrio 8y ago
  We're up to three completely different interpretations of "like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate truth". My takeaway: be wary of analogies; they will be misunderstood.
- myWindoonn 8y ago
  You grokked it. Imagine standing on a vast desert. You have, floating above you, the HYPERCUBE. You could choose to climb up into the HYPERCUBE, but it has no exits. In a serious, non-trivial sense, mathematics is the pattern science. If you choose to believe that reality has underlying patterns, then mathematics is unavoidable. To respond to the grandparent, we can put sets on categories, types on categories, categories on sets, logic on sets, logic on categories, etc. Who cares which foundations you pick, when they all can build from each other?