[FOM] 729: Consistency of Mathematics/1

[W.Taylor at math.canterbury.ac.nz]

Quoting Harvey Friedman <hmflogic at gmail.com>:

> What are the most monumental issues in the foundations of mathematics?

This is a good question, (even if maybe tending to lead to polemics unfriendly
to the functioning of the mailing list).  I will tender a few thoughts,
though hardly original.

> 1. Is mathematics consistent - I.e., free of contradiction?

Certainly an important question, but is it really one that is important
to be "answered", whatever that might mean.  Obviously any discovered
inconsistency will surely lead to a rapid scurrying around of termite
worker mathematicians, to hurriedly seal up the breach in the walls,
as has arguably happened in the past from time to time.  (Though I
suspect the historical cases have never had the clarity that a new
contradiction today would have!  The historical cases are somewhat vague.)

In the absence of a convincing argument for consistency, along with failure to
find an inconsistency, most of us will just continue unperturbed doing math,
and the above will remain a (somewhat pointless?) philosophical quandrary.

> After all,
> it deals with objects not accessible to us by observations in any
> usual sense.

Is there any philosophically-agreed "usual sense"?  And indeed, I am reminded
of a comment by Hardy, that "the chief difference between the mathematician
and the physicist is that the former has a much more direct connection with
reality".  Though put up as a semi-jocular provocation, it is a very
arguable position!

> Not only does mathematics use infinitely many objects, it
> uses myriads of completed infinite totalities.

Indeed so.  But this has small bearing on the previous comment, I feel.
Very large finite numbers have similar objections, after all.

> Can we, and how can we be assured of the consistency of mathematics?

Perhaps more to the point - do we really *meed* to be assured of it?
As long as no contradiction is found, sail on.  If one is found,
patch it up.

> What would constitute
> evidence or confirmation of the consistency of mathematics?

Indeed, a very pertinent and apposite question.  What *kind* of evidence
could there *be* in favour of consistency?  Apart from, as noted,
the continuing failure to find the reverse. (Now surviving since about 1890.)
Is absence of evidence necessarily evidence of absence?

> For those who take abstract set theory as intrinsically fundamentally
> important in its own right, independently of the quality and quantity
> of its connections with the rest of mathematics,

It is quite possible to take set theory, along with its own concerns,
as extremely important, merely for intellectual interest, and without
having to think that they really "Platonically exist" in their full glory.
I think this is sometimes denigrated as the "if-then philosophy of math",
but I fail to see any cause for denigration.  (Mind you, I would not award
PA and its N the same "free-ranging" semantics!)

Indeed I suspect there are few survivors of the other, fully Godelian view.

> 3. Is the continuum hypothesis true or false? Can we and how can we
> obtain evidence or confirmation either way?

On the first point, it is true in some models and false in others, neither
of which has any special claim over the other.  On the second, there
can hardly be any.  (All this being on the if-then view of set theory.)

> Research on 3 could use a refreshing new methodology, which I hope to
> be offering in Refuting the Continuum Hypothesis?/1-9 and hopefully
> continuing.

That will be a good thing to see.

> There are some monumental, or arguably monumental issues that I would
> not classify as foundations of mathematics, bur rather foundations of
> computer science.  ...  ... b) Church's Thesis. Actually b) is such a
> radically different problem than a), that I prefer to view b) as an
> arguably monumental issue in Philosophical Computer Science, which I
> like to distinguish from Foundations of Computer Science, and
> certainly from Computer Science itself.

I go along with all those remarks.

However: there was a recent small thread on the topic, in which various
arguments were put forward and summarized by Tim Chow, that purported to
counter my claim that it is *merely* a matter of definition.  I found
none of the counter-claims at all compelling, reducing AFAICA to mere claims
about what mathematicians/CS-ers might think in the future.  The comparison
with the definitions of continuity and smoothness were noted, very appositely
in my view. I might also adduce the definition of *connectedness* in topology.
I feel it was made "wrongly" (i.e slightly contra to most people's feeling
of the meaning of words (i.e. "intuition")); and that "arcwise connected"
(as it is now known) is what *really* should have been called "connected",
and that what is now called "connected" could be perhaps better called
"unseparable", or somesuch.

But it is obviously a bit silly to argue about what "connected" really means.
In a similar way, I feel it is a bit silly to argue about what computable
*really* means - it is just a definition, with remarkably many seemingly
different characterizations, which all go to show that we have settled
on a VERY USEFUL definition indeed, with limiting cases both above
and below - but a definition nonetheless.

I realize that this is not a popular view, but I cannot understand why not.

> So perhaps we can add
> Philosophical Computer Science to a list of subjects that includes
> Philosophical Geometry - although for Philosophical Geometry I do
> have a reasonably coherent and perhaps resonating view as to just what
> I mean.

Yes, there is a nice study beginning there.

And now perhaps it is time for me to stop, before getting even more plangent!

-- Bill Taylor


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