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> Fortunately, math has an incredibly powerful tool that helps bridge the gap. Namely, when we come up with concepts, we also come up with very explicit symbols

by lutusp 14y ago

> Fortunately, math has an incredibly powerful tool that helps bridge the gap. Namely, when we come up with concepts, we also come up with very explicit symbols and notation, along with logical rules for manipulating them. It's a bit like being handed the technical specifications and diagrams for building a vacuum cleaner out of parts. Just to play the devil's advocate ... math has wonderful symbols and methods, but they don't really assist in comprehension unless everyone agrees on their meaning, and then only if everyone already understands the underlying concepts, the axioms. For example, starting in 1910, Bertrand Russell and Alfred North Whitehead published "Principia Mathematica" (a borrowed title): http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica But, notwithstanding their high intellectual level and the ambitions behind the project, and notwithstanding the system of symbols used, Russell and Whitehead missed a crucial, central point -- their plan to systematize mathematics, place it on a solid logical foundation, make it immune from uncertainty and doubt, was doomed from the start. Kurt Gödel demonstrated this a few years later: http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theorems http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo... So much for the power of symbols. The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds.

cwzwarich 14y ago
> The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds. The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. They also don't remove the possibility that there is some metamathematical justification for an unambiguous interpretation of the concept of the natural numbers or even of a set (see all of the work in large cardinals, culminating in Hugh Woodin's recent work on extender models for supercompact cardinals).
- lutusp 14y ago
  > The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were. Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis. > They also don't remove the possibility that there is some metamathematical justification for ... Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.

by lutusp 14y ago

cwzwarich 14y ago

> The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds. The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. They also don't remove the possibility that there is some metamathematical justification for an unambiguous interpretation of the concept of the natural numbers or even of a set (see all of the work in large cardinals, culminating in Hugh Woodin's recent work on extender models for supercompact cardinals).

lutusp 14y ago
> The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were. Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis. > They also don't remove the possibility that there is some metamathematical justification for ... Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.