[FOM] Fact and opinion in F.O.M.

[Joe Shipman]

I distinguish between a “fact” and a “tautology”, although if you make a good enough argument for logicism I would be willing to reclassify mathematical theorems as tautologies rather than facts.

And no, I clearly identified the phenomenon that many results in formally infinitary subjects such as topology and knot theory can be shown (in weak systems) to be equivalent to arithmetical statements. It is only the results of topology that can NOT be shown to be equivalent to arithmetical statements that I have doubts about the facticity of (and the particular subtheory of  topology known as “knot theory”, thanks to the efforts of Moise and others, has already been shown equivalent to its piecewise linear analogue which can be straightforwardly coded into finite mathematics).

— JS

Sent from my iPhone

> On Dec 27, 2019, at 1:20 PM, Michael Lee Finney <michael.finney at metachaos.net> wrote:
> 
> 
>> Joe Shipman wrote:
> 
>> Then your question becomes whether there exist any non-arithmetical facts.
> 
> I am puzzled as to why this is even a question. What about results in
> Topology, or Knot Theory? Or in logic. Surely "p & q -> p" should be
> considered a "fact"? None of these are arithmetical consequences.
> 
> And if the criteria for a fact is that no set of axioms exclude it, then are
> there really any facts at all?
> 
> Even if the above examples are modeled in ZFC, that does not make them a
> consequence of ZFC. At best you could say they are a consequence of ZFC plus
> additional axioms. Since you could negate any one or all of those additional
> axioms or an axiom of ZFC itself, you would always have a potentially proposed
> system where any given statement is false.
> 
> I don't think you can restrict any proposed set of axioms based on
> "agreement", "seriously proposed", etc. That criteria just shifts in the wind.
> Just as with non-Euclidian geometry, there was no "permanent disagreement" for
> hundreds of years -- and then there was -- which, over time, changed to a
> new agreement.
> 
> Nor can you restrict it to arithmetical facts, or systems which can be
> modelled in ZFC or even just ZF. What about set theories which replace the
> axiom of regularity (such as Azcel's systems)? Some of those have been shown
> to be consistent if and only if ZFC is consistent. And they contain "facts"
> that disagree with ZFC. And those systems have been seriously proposed as
> replacements for ZFC.
> 
> At best you can say that a "fact" is something that follows from a specific
> set of axioms. And an "opinion" is a belief that something "should" be true,
> even if it is independent of that set of axioms, such as the axiom of choice.
> Facts and opinions in political discourse are an entirely different kettle of
> fish. Almost all "facts" in that arena are really opinions -- often very
> strongly held opinions -- but nonetheless opinions and not facts.
> 
> I would also argue that people of good will can and are in permanent
> disagreement about "facts". The gun control "debate" is one such area. What is
> a fact for one side is viewed as an incorrect or ignorant opinion by the other
> side (not taking a side here). There are many areas like this -- especially
> where public policy is concerned -- where people of good will have
> irreconcilable differences about the "facts" of the debate.
> 
> Michael Lee Finney
> 
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