
Harvey Friedman bringing incompleteness and infinity out of quarantine - dnetesn
http://nautil.us/issue/45/power/this-man-is-about-to-blow-up-mathematics
======
fmap
There is a lot of ordinary mathematics that is outside of ZFC, which Friedman
is well aware of but the writer of this article may not be. Grothendieck was
not interested in abstract set theory when he introduced what are now called
"Grothendieck Universes". He merely wanted to do algebraic geometry at a high
level of abstraction. Similarly, Conway was apparently a bit dismayed when the
formalization of his simple idea of "surreal numbers" required deeply
unnatural encodings in ZFC...

My opinion is still that ZFC itself is unnatural as a foundation for
mathematics, precisely because we have to do so much encoding to get anything
useful out of it. And whenever you iterate "large" encodings you leave the
universe of "ordinary ZFC". Conway suggested - and this is realized in modern
type theory - that we should instead allow arbitrary "free" constructions,
such as his surreal numbers, to extend the basic universe of mathematics. To
some extend this can be encoded in set theory, but only with ridiculously
large (Mahlo) cardinals.

This is not a direction that Friedman considers worthwhile, because he thinks
that first-order logic and ZFC are inevitable. It's a shame that so many
people on the FOM mailing list share the same view. There are good reasons why
you don't find a lot of category theorists on that list anymore...

~~~
pron
> a lot of ordinary mathematics that is outside of ZFC

 _A lot_ of _ordinary_ math? I think there's one if not two exaggerations in
there.

> My opinion is still that ZFC itself is unnatural as a foundation for
> mathematics, precisely because we have to do so much encoding to get
> anything useful out of it.

One of the things Friedman likes to emphasize about the foundation of math is
that there is very little you actually need to do with it. A good foundation
is one that you don't even need to know is there. Most mathematicians don't
work in any formalism, so a foundation doesn't need to be useful. It's not as
if anyone actually needs to _do_ all those tedious set encodings.

 _Except_ that many of those who call for new foundations are really
interested in mechanical proof checkers, and those do have to be useful. In
order to make these arguments less religious, I think it would be worthwhile
to adopt Turing's mathematical philosophy, which called for adopting
mathematical foundations on an ad-hoc basis (or even no foundation at all). In
other words, choose whatever foundation (if any) for the task at hand. This
would make it easier to argue that, say, type theory is a more convenient core
for proof checkers.

> This is not a direction that Friedman considers worthwhile, because he
> thinks that first-order logic and ZFC are inevitable.

That's not how I read him. He thinks that FOL and ZFC are good enough, and as
foundations don't matter in practice (a good foundation is one you can ignore;
I think that's a quote by him), it would take some achievement to justify
seriously considering alternatives. He even says what it would take: easier
teaching and/or easier proofs. So far no alternative foundation has been able
to improve either one.

~~~
eastWestMath
>In order to make these arguments less religious, I think it would be
worthwhile to adopt Turing's mathematical philosophy, which called for
adopting mathematical foundations on an ad-hoc basis (or even no foundation at
all). In other words, choose whatever foundation (if any) for the task at
hand. This would make it easier to argue that, say, type theory is a more
convenient core for proof checkers.

I think you're misrepresenting the category theorists, they're usually the
ones arguing for choosing whatever foundation is convenient for a given
domain. A lot of the time, this is a type theoretic foundation because a lot
of modern mathematics is about pretending you have function types when you
don't actually have function types in your category (such as differential
geometry) or that all you care about is a "core" set of operations and the
rest of the framework you're working in simply gets in the way (such as
Hilbert spaces vs. compact closed dagger categories in categorical quantum
mechanics). And if you know already knew dependent type theory then synthetic
homotopy theory _is_ easier than classical homotopy theory, some proofs in
homotopy theory can take several lectures to present and even then it will
still be pretty hard to see why they hold.

It's just weird to see people thing the category theorists are the one being
impractical compared to the set theorists like Friedman. I can't think of a
categorical logician who doesn't have an active line of research outside of
logic; usually in topology, computability, quantum mechanics, or algebraic
geometry. They're not just logicians but active researchers in computer
science, physics and mathematics, logic is a powerful tool and _should_ be
applied in these fields.

~~~
pron
I'm not trying to rule who is more reasonable (and frankly, I have no idea)
and certainly not to misrepresent category theorists (whom I didn't mention),
but just respond to fmap's statement that "ZFC itself is unnatural as a
foundation for mathematics", which assumes that 1. a foundation is something
is regularly used for _anything_ , and 2. that it is something that is both
fixed and essential for mathematics; in other words that the name
"foundation", rather than signifying the study of math at the lowest levels,
actually refers to something fundamental mathematics "rests on". Very little
math is done on top of a formal foundation at all, and most of math does not
even require a foundation. In fact, it is questionable whether any formal
"foundations" are really foundational, or, as Wittgenstein called them, "just
another calculus". If that is the case, it's meaningless to argue that ZFC
isn't a natural foundation for mathematics, because in spite of the name, it
may not really be a foundation at all; just another calculus that is called
"foundational" because of its level of discourse, in which case the arguments
against it can only be aesthetic or pragmatic. In order to be pragmatic, you
need to say what for what uses the foundation is inadequate.

But I think it's naive to pretend that there's no fight over aesthetics here,
which is why I mentioned Turing, as I see a return to logicistic (as in
logicism) arguments (which view mathematical foundations as truly fundamental)
in new form, while Turing's philosophy manages to, according to Wittgenstein,
avoid "needless dogmatism and dispute". I strongly encourage anyone interested
in the subject to read Juliet Floyd's fascinating paper on Turing's
mathematical philosophy:
[https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf](https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf)

------
jesuslop
If you want to hear of Friedman in real debate there is great discussion in
the foundations of mathematics mailing list archives that are public. There
there is real lively yet high-standards scholar figth of first rate experts
from all viewpoints. I liked for instance the 'myth of second order logic'
theme initiated by S. Simpson.

It seems to me that the article is wrong when it says that the spheres
recompounded bigger are a problem of large cardinal axioms. I think it derives
from Choice, the C in ZFC instead.

~~~
golergka
> I liked for instance the 'myth of second order logic' theme initiated by S.
> Simpson.

Can you link to the thread? The archives seem gigantic

~~~
wfn
Possibly
[http://www.personal.psu.edu/t20/fom/postings/9903/msg00069.h...](http://www.personal.psu.edu/t20/fom/postings/9903/msg00069.html)
(and Simpson's replies downthread, e.g.
[http://www.personal.psu.edu/t20/fom/postings/9903/msg00151.h...](http://www.personal.psu.edu/t20/fom/postings/9903/msg00151.html))

------
lmm
> Showing it’s not provable, on the other hand, is more difficult. He did this
> with a proof by contradiction: He began with the assumption that he could
> prove his theorem in ZFC, and then constructed from it a system of objects
> in which ZFC holds. Which means that if his theorem holds true, then ZFC is
> consistent—and, transitively, that ZFC has proven its own consistency. But
> by Gödel’s incompleteness theorem, that cannot possibly be the case. And so,
> the theorem cannot be proven in ZFC. He’s working to extend the theory to
> other types of symmetries, other definitions of “maximal,” and other types
> of objects.

So he encoded some statements about ZFC into these statements about sets of
rationals. Wasn't Gödel was able to do better already, encoding statements
about ZFC into basic arithmetic/number theory? I guess I just don't understand
what the big breakthrough is supposed to be, even though I'm interested in
alternate axiom systems (e.g. the whole homotopy type theory / univalent
foundations business). Is the idea that he's come up with some new natural
statements about symmetries of sets that turn out to demonstrate
incompleteness? That would be an innovation, but the above description makes
it sounds like it's more about finding symmetry statements that correspond to
ZFC.

------
Mathnerd314
His recent posts on equivalence theory:
[http://www.cs.nyu.edu/pipermail/fom/2017-February/020299.htm...](http://www.cs.nyu.edu/pipermail/fom/2017-February/020299.html)
[http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.htm...](http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.html)
[http://www.cs.nyu.edu/pipermail/fom/2017-February/020301.htm...](http://www.cs.nyu.edu/pipermail/fom/2017-February/020301.html)

They still don't strike me as particularly natural or simple, particularly
when compared to my favorite example of incompleteness, the surreal numbers.
The surreal numbers are defined via transfinite induction and can grow as
large as whatever cardinal you choose, and there are good reasons to pick
large cardinals (closure under various operations), with the associated
complications.

> “The idea that there’s absolute solidity, a right and wrong, in
> mathematics—that mathematics has no real conceptual philosophical issues
> that have to be dealt with … I’m interested in completely blowing that up.”

First-order logic is sound and complete; it's only second-order and higher
logic that supports Godel's incompleteness theorem. Since most math is done in
first-order logic (plain ZF, not even C; I think this is what the 85% figure
in the article is referring to) I'm pretty sure this is not going to happen.

~~~
cousin_it
ZF is subject to the incompleteness theorems, it can't prove Con(ZF). Even
much simpler systems, like Peano arithmetic, are subject to them.

~~~
Mathnerd314
It's consistent/complete in first-order logic, because Con(ZF) is a higher-
order statement and thus can't even be formulated.

First-order logic has a lot of nice properties, e.g. "if a result is finitary
in the sense that it can be phrased as a first-order statement in Peano
Arithmetic, and it can be proven using the axiom of choice (or more precisely
in ZFC set theory), then it can also be proven without the axiom of choice
(i.e. in ZF set theory)." (c.f.
[https://terrytao.wordpress.com/2013/12/07/ultraproducts-
as-a...](https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-
between-discrete-and-continuous-analysis/))

~~~
cousin_it
Con(ZF) is a sentence in ZF.

------
orbifold
I think homotopy type theory has a better shot at providing a good foundation
for math as opposed to some revision of set theory.

~~~
AnimalMuppet
I don't understand HoTT. But every time I try to learn a bit about it, I go to
the Wikipedia article (hey, everybody's got to start somewhere). And something
always bugs me:

They're trying to build a foundation for all of mathematics. But their
starting point requires weak omega groupoids. But weak omega groupoids are not
exactly a fundamental object; you kind of need a foundation of something else
before you ever get anywhere near weak omega groupoids. Their foundation,
then, seems to me to be inevitably recursive.

Now, I get that the people involved in this are much smarter than me, and know
far more mathematics. So, if you can, ELI5: Why isn't HoTT a recursive house
of cards?

~~~
Kutta
HoTT doesn't build on any notion of omega groupoids; at the lowest level it's
a collection of typing and computation rules which one can apply successively
to do proofs and constructions. The resulting constructions can be interpreted
as talking about properties of spaces. The rules themselves are very stripped-
down and abstract when viewed in a homotopical light, like "every path can be
retracted to an endpoint" or "the interval has two points and a path between
them". Originally the basic rule for reasoning about paths ("path induction")
was intended to allow construction of equality proofs between elements of
types. Later people discovered that equalities can be interpreted as paths in
spaces (and equalities of equalities as homotopies and so on).

Remarkably, the four typing rules of equality suffice to generate all
homotopical reasoning in classic HoTT. However, they aren't enough to prove
univalence as a theorem, or at least no one knows how to do it. If we switch
to cubical type theory, we get considerably more structure which allows us to
prove univalence. But cubical type theory is also "synthetic" and builds up
notions of spaces from ground-up.

~~~
AnimalMuppet
Thanks to you (and Chinjut) for the replies. I don't know how to correlate
your reply to Chinjut's, though (or vice versa). Could either of you take a
stab at it? Are you saying the same thing in different ways? Or are you
actually disagreeing?

~~~
Kutta
The same thing as far as I see.

~~~
AnimalMuppet
So when you said, "at the lowest level it's a collection of typing and
computation rules which one can apply successively to do proofs and
constructions", That was what Chinjut meant by "Homotopy Type Theory
axiomatizes weak omega groupoids; it gives you formal rules you can manipulate
to reason about weak omega groupoids"? That is, those lowest-level rules don't
_assume_ weak omega groupoids; they turn out to _define_ weak omega groupoids?

------
arketyp
How does this relate to automata theory and Wolfram's principle of
computational equivalence (spare me any lectures about his self-promotion) and
the conjecture that universal Turing machines are virtually ubiquitous and
arise from the simplest of rules?

------
ibgib
Interesting article. I can relate to his dictionary investigations when I was
confronted with being taught about "circular definitions". I couldn't get
around (no pun intended) the fact that _all_ definitions were circular.

This is what has led me to developing ibGib, which as I stated here elsewhere
([https://news.ycombinator.com/item?id=13632500](https://news.ycombinator.com/item?id=13632500))
that ibGib is heavily influenced by Goedel in that it uses a SHA-256 hash of
an immutable datum to effectively be the Goedelian number of that datum. So
there are four fields: ib, gib, data, rel8ns. The gib is the hash of the other
three fields, which allows for integrity of the data, as well as allowing it
to be content addressable since the ib^gib is the address. So it's a
monotonically increasing data store that focuses on the _process_ of defining.

So instead of "proofs" that imply an ontological/objective "truth", ibGib
focuses on "the meta proof" that is the prov- _ing_ process by assigning any
"proving mechanism" a number (just as it assigns _anything_ a number). This
way, you are approaching paradoxes and the like as a process of _economics and
evolution_ of proving systems. This is like the "proof" _that contains the
peer review process that is doing the proving_.

------
JumpCrisscross
> _Given any class of mutually exclusive classes, of which none is null, there
> is at least one class which has exactly one term in common with each of the
> given classes..._

If anyone else's train of thought got wrecked by the proposition of mutually-
exclusive classes sharing terms, here you go:
[http://www.encyclopedia.com/people/science-and-
technology/ma...](http://www.encyclopedia.com/people/science-and-
technology/mathematics-biographies/bertrand-arthur-william-russell).

~~~
rntz
The "there is at least one class which.." doesn't mean "there is at least one
class _in that class of mutually exclusive classes_ which..."; it means "there
exists at least one class _at all_ which...". This is one way of phrasing the
axiom of choice (although Russell uses "class" instead of "set"; in modern-day
usage, we distinguish classes from sets, and AC applies to sets only).

~~~
JumpCrisscross
It made sense once I mentally substituted "there _exists_ at least one class"
for "there _is_ at least one class".

------
d3ckard
It's especially interesting given that P vs NP is suspected to be unprovable
in ZFC. Waiting for further development!

~~~
logicallee
I'm curious why you say "P vs NP is suspected to be unprovable in ZFC".

What would make mathematicians suspect that (if you know)? I mean ZFC is not a
toy system, it is obviously used in complex and deep ways to prove or resolve
long-standing questions in ways that build on tons of deep results that
themselves took tons of research. (By the way for anyone else reading, just so
you don't get the wrong idea: ZFC is actually the standard set of axioms
mathematicians work with every day. It's the normal way to do math.)

What would make someone suspect this one is impossible to prove/decide under
ZFC? (Or what would make someone think it is independent under ZFC?)

"A lot of people have tried to prove it" doesn't seem a convincing argument.
(Consider advances toward the twin prime conjecture, or the now-proven
Fermat's last theorem, etc.) There must be something more that makes you say
that.

Thank you for any answer.

~~~
rocqua
> ZFC is actually the standard set of axioms mathematicians work with every
> day

I'd dispute that mathematicians work with ZFC every day. Most pure but not
foundational mathematics basically works with intuitionist set theory. By
keeping the size and nesting of sets small enough, we never need to worry
about ZFC.

Zorn's lemma comes up occasionally, but that is all.

~~~
logicallee
Wikipedia says "Today ZFC is the standard form of axiomatic set theory and as
such is the most common foundation of mathematics."

What my parenthetical comment that you quoted meant to say (for anyone that
was not familiar with it) is thaz ZFC is not some joke, or obscure set of
axioms or something irrelevant. Without this remark I thought my comment read
that way.

I hope you will agree that ZFC is simply "standard math" \- even if its axioms
are not referenced explicitly by many working mathematicians.

If this statement requires further refinement let me know. Obviously I'm not a
mathematician!

~~~
rocqua
Certainly, ZFC is not a pathological constructed example. When a mathematician
looks for a foundational set of axioms, ZFC is THE standard choice. It is, and
has been, for the last ~90 years at least.

My point was that foundational mathematics is rarely touched upon by a lot of
normal pure mathematics (say number theory, field extension, graph theory).

Interestingly, I believe a lot of people actually dislike the axiom of choice.
They find it to be way to 'complex' when compared to the other axioms. It is a
lot like euler's 5-th axiom (2 straight lines with the same direction are
equidistant everywhere). Interestingly in that case, removing the axiom led to
spherical and hyperbolic geometry.

~~~
benibela
You could say the mathematicians work as much with ZFC as programmers work
with assembly

