[FOM] Matters of Fact

[Harvey Friedman]

Some correspondence between John Baez and me. Below see first John,
and then my reply I just sent to

Roux Cody
on Foundations of Mathematics (mailing list)
google+

John Baez wrote:

I don't know what it would mean to assert that some mathematical
axioms are "true" and others are "false", unless one is doing
something like physics and making the additional extra-mathematical
claim that the rules are obeyed by something we can study by
experiment.

I am in some ways a Platonist, in that I think mathematical patterns
underlie the universe and have a type of existence of their own: one
might call it "potential" existence, in that when the necessary
conditions for a pattern's existence are met, that pattern gains sway
over what occurs.

For example, if water obeys some sort of wave equation, the
consequences of that equation occur, while if the water becomes cold
and minimizes its energy, a crystal lattice will appear.  If we agree
on the rules of chess, the opening "pawn to king 4" becomes typical
among winning chess players.  If we choose to prove theorems starting
from the axioms of ZFC, certain things become inevitable.  If we
choose HoTT, other things become inevitable.

However, some rules are more interesting than others, and it's one of
the highest expressions of mathematical taste to be able to choose
interesting rules to work with.  There are lots of constraints on what
count as interesting rules... and yet, still, much freedom remains.

Friedman wrote:

I feel like we are dancing around the really crucial issues in the
foundations of mathematics. We are not quite there yet in terms of
real engagement on this site and others.

The attitudes toward these crucial issues are reflected in the
perceived levels of relevance or shock value in Incompleteness, and
play a vital role in the evaluation of and construction of research
programs in Incompleteness.

I raised a few issues in my April 3, 2016, and I'm not quite sure what
your 3:40PM "Very roughly, yes" refers to. Let's consider three kinds
of mathematical statements.

1. Statements like CH, which involve unrestricted uncountable sets.
2. Statements like Twin Prime Conjecture, Goldbach's Conjecture, or
e+pi is irrational, which (can be set up to) live in the ring of
integers.
3. Statements like 2 only restricted to integers of magnitude at most
2^2^...^2, where the stack has height 2^2^2^2^2^2^2.

Now I was thinking that those inclined to like various kinds of
categorical foundations definitely feel that in category 1, there is
no matter of fact, and there is only what system you wish to set up
(say topoi and the like), where there are no preferred setups, or
there are many more or less equally preferred setups, and these give
different truth values to the statements. I think you might have meant
"Very roughly, yes" to that.

A consequence of that position is that there is no monumental
philosophical foundational revelation in the Goedel/Cohen results that
CH is neither provable nor refutable in ZFC. That this definitely does
not change the way your categorically oriented friends look at the
very fundamental nature of mathematics. That you do not find the
Goedel/Cohen results deeply disturbing, challenging long held beliefs
in the crystal clear objective nature of the ONE mathematical reality.
Etc.

Before doing so, with regard to 1. The vast majority of set theorists
have spent decades taking the point of view that statements in 1 are
matter of fact, and there is only one set theoretic reality. This has
been softened some in various in recent years. One softening is not so
severe. This softening says that OK, mathematical objects don't form a
Platonic reality, but there is a completely objective matter of
factness involving statements about mathematical objects, in
particular the universe of sets. This is a tricky dance to make
coherent, but nevertheless it is out there, as a way to still maintain
that statements like CH are matter of fact.

But there has also been a growing movement, even among set theorists,
that there is not just one "universe of all sets", and so statements
like CH are not matter of fact. This movement would have been
unthinkable in set theory even a decade or two ago.

I'm not sure that the typical working mathematician has any opinion as
to whether statements like CH are matter of fact or not. My best guess
is that when it becomes clear to them that this involves *arbitrary*
sets of real numbers, which generally have no numerical,
computational, or geometric significance, they will back off from any
initial impulse to say that statements like CH are matter of fact.

It could also be the case that the working mathematician, particularly
maybe the young ones, are kind of implicitly trained not to think
about any philosophical or foundational questions at all, considering
such things a total waste of time, and possibly damaging to their
career?

What about mathematical logicians? I would venture to say that a
significant minority of them still believe rather unequivocally that
statements like the CH are matter of fact. But the majority, not
overwhelming majority, is definitely at least uncomfortable now with
this view - especially since there has been the beginnings of a
rebellion in the set theory community.

NOW there are quite of number of steps along the way from 1 and 2
above. And that also can generate some important foundational
discussions. But I want to move to 2.

2. Statements like Twin Prime Conjecture, Goldbach's Conjecture, or e+pi

Now here, the preponderance of mathematical logicians think that
statements of kind 2 are indeed matter of fact. They are fully aware
that there is not such a strong analogy with experimental science
here. The closest would be regarding Goldbach's Conjecture, where in
principle the statement could be refuted by example. Of course even
here, maybe the counterexample is too big for a computer to deal with
it directly, but the counterexample may be something subject to a
combination of theory and maybe some computation, thereby refuting
Goldbach's Conjecture.

For the working mathematician, I think also that the majority of those
willing to think about foundational issues, statements of kind 2 are
matter of fact. The working mathematician might or might not be
sensitive to an importance difference between a statement like the
Twin Prime Conjecture and a statement like Goldbach's Conjecture. The
former is a "for all there exist" statement and the later is a "for
all" statement. Only the latter is subject to a thought experimental
refutation.

Now lets move all the way down to 3. Here I think that nearly all
mathematicians think that there is a matter of fact about such
statements. That there is no such thing as an alternative mathematical
universes when it comes to statements like 3.

With 3, the numbers too big for (direct) experimentation, however. And
so there are various levels below 3 that are very foundationally
interesting.

As far as I can tell, it is only the levels approaching 3 for which
these arguments about foundational frameworks, of the kind here, make
any difference whatsoever.

Where does John Baez' thinking fit into this picture.

Harvey Friedman