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> Gödel quite literally proved mathematics is "beyond" (as in its truths cannot be wholly captured by) and "above" (as in mathematical rules seem to be able to

> Gödel quite literally proved mathematics is "beyond" (as in its truths cannot be wholly captured by) and "above" (as in mathematical rules seem to be able to characterize the whole of) mathematical logic. Godel's Incompletness Theorems do not by themselves address truth, but instead require additional theorems and/or philosophical commitments to decide either way.

morelisp 6y ago
> Godel's Incompletness Theorems do not by themselves address truth This is itself a sort of philosophical commitment - and again, not one shared by Gödel or the logicist program.
- dwohnitmok 6y ago
  I mean yes, in so far as any statement about philosophy is a philosophical commitment, but that kind of feels like nitpicking no? In very real ways Godel's Incompleteness Theorems do not address truth. Their proofs are purely syntactic rather than semantic. > and again, not one shared by Gödel or the logicist program. The logicist program is a bit tricky to pin down; the major figures associated with it either were no longer active in mathematical logic by the time of Godel's result (e.g. Russell) or were but never made any response to them (e.g. Hilbert). However, we know Godel's take on the matter. In particular I don't think this is an accurate account of Godel's own viewpoint. It is clear Godel clearly believed that there were mathematical truths that could not be expressed by formal systems. However, it is also clear that Godel acknowledges this to be an additional philosophical commitment. Godel himself cleanly states this in his 1951 Gibbs Lecture. > Does it [his incompleteness theorems, in particular his second one] mean that no well-defined system of mathematics can contain all of mathematics proper? It does, if by mathematics proper is understood the system of all true mathematical propositions; it does not, however, if one understands by it the system of all demonstrable mathematical propositions. I shall distinguish between these two meanings of mathematics as mathematics in the objective and in the subjective sense. Godel is pointing out that his incompleteness theorems require an additional philosophical commitment to "objective" or "subjective" mathematics in order to make a statement on truth (what he refers to as "true" in his comment already demonstrates his own predilection towards "objective" mathematics being the correct philosophical commitment, but the important point here is that he is making the distinction). As a footnote, Godel's notion of mathematical truth is an extremely strong Platonist position that I don't think most HN commentators would agree with (at least at the beginning; they may be swayed to come around to it). Godel believed there were no truly independent mathematical statements, only ones temporarily independent of our limited set of axioms. Every mathematical statement is either true or false. For example, the continuum hypothesis will be definitively decided one way or the other (if I recall Godel ultimately estimated the continuum hypothesis to be false).

morelisp 6y ago

> Godel's Incompletness Theorems do not by themselves address truth This is itself a sort of philosophical commitment - and again, not one shared by Gödel or the logicist program.

dwohnitmok 6y ago
I mean yes, in so far as any statement about philosophy is a philosophical commitment, but that kind of feels like nitpicking no? In very real ways Godel's Incompleteness Theorems do not address truth. Their proofs are purely syntactic rather than semantic. > and again, not one shared by Gödel or the logicist program. The logicist program is a bit tricky to pin down; the major figures associated with it either were no longer active in mathematical logic by the time of Godel's result (e.g. Russell) or were but never made any response to them (e.g. Hilbert). However, we know Godel's take on the matter. In particular I don't think this is an accurate account of Godel's own viewpoint. It is clear Godel clearly believed that there were mathematical truths that could not be expressed by formal systems. However, it is also clear that Godel acknowledges this to be an additional philosophical commitment. Godel himself cleanly states this in his 1951 Gibbs Lecture. > Does it [his incompleteness theorems, in particular his second one] mean that no well-defined system of mathematics can contain all of mathematics proper? It does, if by mathematics proper is understood the system of all true mathematical propositions; it does not, however, if one understands by it the system of all demonstrable mathematical propositions. I shall distinguish between these two meanings of mathematics as mathematics in the objective and in the subjective sense. Godel is pointing out that his incompleteness theorems require an additional philosophical commitment to "objective" or "subjective" mathematics in order to make a statement on truth (what he refers to as "true" in his comment already demonstrates his own predilection towards "objective" mathematics being the correct philosophical commitment, but the important point here is that he is making the distinction). As a footnote, Godel's notion of mathematical truth is an extremely strong Platonist position that I don't think most HN commentators would agree with (at least at the beginning; they may be swayed to come around to it). Godel believed there were no truly independent mathematical statements, only ones temporarily independent of our limited set of axioms. Every mathematical statement is either true or false. For example, the continuum hypothesis will be definitively decided one way or the other (if I recall Godel ultimately estimated the continuum hypothesis to be false).