[FOM] Alternative Foundations/philosophical

Nick Nielsen john.n.nielsen at gmail.com
Wed Feb 26 21:15:05 EST 2014

The "Mathematicians' Liberation Movement" characterized by Chow might
be formulated in terms of the Principle of Tolerance, or, to be more
specific, the Principle of Tolerance in Foundations. Precisely
parallel to Carnap, one can say that it is not our business to set up
prohibitions, but to arrive at conventions in FOM.

Chow's advice to, "open-mindedly explore if the new system helps
foster new ideas," is an instance of what Carnap called, "simultaneous
investigation... of language-forms of different kinds."

Carnap famously said, "In logic, there are no morals. Everyone is at
liberty to build up his own logic, i.e., his own form of language, as
he wishes. All that is required of him is that, if he wishes to
discuss it, he must state his methods clearly, and give syntactical
rules instead of philosophical arguments."

Here Carnap lays down a condition of syntactical rules, but in a
Principle of Tolerance in Foundations we could just as well lay down
the conditions that Harvey Friedman has formulated, and say that,
"Everyone is free to elaborate his own foundations. All that is
required of FOM is that its discussion embody absolute rigor,
transparency, philosophical coherence, and addresses fundamental
methodological issues."

Of course, cashing out these conditions will be a lot more difficult
than laying them down.

Best wishes,

Nick Nielsen

On Tue, Feb 25, 2014 at 12:43 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
> I agree with a great deal of what Chow has just said. However, I have some
> problem with a full endorsement of the following from his message:
> \
> "I am sympathetic to John Conway's "Mathematicians' Liberation Movement"
> (which of course has also been discussed before on FOM), which basically
> says that we're mature enough now to be able to pick whatever foundations
> we find convenient, knowing that we can always, in principle, translate
> between any two if them if we really want to.
> In other words, if a proponent of a new system claims certain advantages
> over the old system, I do not think the reaction should be to get all
> defensive and say, "But I can do that with the old system too!"  Instead,
> one should open-mindedly explore if the new system helps foster new ideas
> that advance the field.  The sooner we abandon childish turf wars, the
> faster mathematics (and the foundations of mathematics) will advance."
> There are two aspects of the usual foundations that are generally accepted
> (or are they generally accepted?).
> 1. There is a crucial kind of absolute rigor in the presentation.
> 2. There is a completely transparent elementary character that is relatively
> universally understandable.
> 3. There is a precious kind of philosophical coherence that transcends
> mathematics itself.
> 4. It has been used in order to address the obvious great fundamental
> methodological issues, the most well known of which concern whether or not
> propositions can be proved or refuted - both generally and specifically.
> A certain amount of this would also be a priori clear for an alternative
> foundation if that alternative foundation was in an appropriate sense
> interpretable in the usual foundation. However, such an interpretation is
> generally not nearly enough to ensure 2.
> How do 1-4 fare with alternative foundations?
> With regard to the "liberation Movement", if one is concerned with fully
> complete rigorous presentations, then has history shown that generally
> speaking one either doesn't have this at all, or one has it done
> incorrectly, replete with inconsistencies?
> Isn't an example of this kind of thing, the idea of using general category
> theory as an alternative foundation, with the "liberated" use of things like
> the category of all categories? Hasn't that been recently shown to lead to
> convincing inconsistencies within the usual mindset of general category
> theory?
> Also, has there been a philosophically coherent presentation of altered
> notions of equality? If so, the FOM would benefit greatly from seeing this
> discussed.
> Harvey Friedman
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