
The Foundations of Mathematics (2007) [pdf] - lainon
https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf
======
fmap
It's surprising that this was written in 2007. This article just perpetuates
the tired old myth that there is something special about set theory... At its
heart, set theory allows us to _encode_ certain mathematical structures more
or less naturally, but things are not "made out of sets".

For example, the real numbers are not sets in the same sense that "the map is
not the territory". Sets allow us to encode real numbers, but this encoding is
mostly arbitrary. Is 3 an element of pi? If pi truly _is_ a set, then this is
a sensible question to ask, since sets are defined by their members. However,
there is evidently an abstract concept of "real numbers" which exists without
mentioning the elements of a real number. The technical reflection of this
dilemma is the fact that in set theory there are multiple isomorphic copies of
"the complete archimedean ordered field" which are all _different_ as sets.

This is one reason why type theory is superior to set theory. In type theory
we can actually capture the abstract concept of real number and work with it,
instead of always working with an encoding. This might not matter so much for
something as simple as a real number, but it matters a lot when you talk about
more complicated structures.

~~~
pron
While I completely agree that set theory is not special nor _the_ foundation
of mathematics (there is no such thing; foundation of mathematics is a name
given to any formal language that can express all or most of mathematics), you
are also repeating a common misrepresentation of set theory. While it is true
that people often present, say, the natural numbers as some arbitrary encoding
in sets, that is not how the natural numbers must be represented in set
theory, and set theory can nicely capture the idea of an abstract inductive
definition of the natural numbers, as well as the isomorphism between
different representation. Many set theories employ Hilbert's epsilon (choice)
operator, that allows one to "choose" some set that satisfies a
proposition[1]. The question of the members of the set is completely
unanswerable, and so can be reasonably said to be nonexistent. You do not work
with a particular encoding, and you can't even if you wanted to, because the
choice operator does not "reveal" what it is.

It is true that the fact that "there exists" (in a very nonconstructive sense)
some unknowable, impenetrable encoding may offend the aesthetical
sensibilities of some people, but type theory does not really "resolve" that
in any way (and it is certainly not superior, just different, or rather, it is
superior in some ways and inferior in others[2]); it simply constructs its
axioms at a higher level of abstraction. The axiomatic existence of inductive
data types in type theory is just a theorem about well-founded sets in set
theory, but those sets are really defined in exactly the same abstract way.

[1]: So the natural numbers can be _defined_ as "epsilon set N, s.t. N is the
minimal set s.t. an element zero exists in N and there exists a function succ
in the set N -> N, such that for all n in N, succ(n) is in n, and there is no
n in N s.t. succ(n) = zero".

[2]: For example, because most type theories are based on the lambda calculus,
they are on the precipice of inconsistency due to Curry's paradox, which makes
some aspects of working with them extremely unpleasant compared to set theory.

~~~
wastewaste
> epsilon set N, s.t. N is the minimal set s.t. an element zero exists in N
> and there exists a function succ in the set N -> N,

this is recursive, which is just how a function like succ works (with a bit
more work, anyway) but the definition rather concernes a polymorph function.

You do work in a particular encoding when you mention yero, at least. and
succ^x is also an encoding (where x is zero or succ^y, where y is zero or
...).

~~~
pron
> You do work in a particular encoding when you mention yero, at least.

That encoding is completely opaque, though. You have no way of knowing what
are the members of zero.

------
unclesaamm
This discussion seems like an apt place to drop Morris Kline's "Mathematics:
The Loss of Certainty" ([https://www.amazon.com/Mathematics-Loss-Certainty-
Oxford-Pap...](https://www.amazon.com/Mathematics-Loss-Certainty-Oxford-
Paperbacks/dp/0195030850)). As a non-mathematician whose only exposure with a
lot of these ideas is through that book, I can't help but see these
discussions as falling very much in line with the competing schools of thought
described in the book.

The book is a non-academic tour of the history of mathematical foundations,
and the way mathematicians struggled to rediscover "truth" or the purpose of
their work when new crises were reached. For example, the book spends a good
deal of time explaining the centrality of Euclidean geometry to people's
worldviews, and the way that the discovery of non-Euclidean geometries shook
people. Not just because they didn't assume other geometries could exist, but
because people believed geometry to map onto Euclidean physical reality
because it was God's way of revealing Himself to the world.

The other main crises that the book toured were the discovery of quaternions,
Cantor's theories, and Godel's theorem.

Kline ends describing the arc over the last two hundred years of math as a
splitting-off into four different schools: set theorists, intuitionists,
formalists, and logicists. Each camp tried to reassert math on "solid ground".
I hear echos of those debates in this thread, where some are asserting that
there can possibly exist multiple foundations, which from my reading of the
book is a very formalist idea (our rules of math are a formal system, and any
internally consistent set of rules are just as valid as objects of study).

Not being a mathematician, I don't have a sense of where those schools played
out to the current day. I'd be curious to hear if they're all still around in
different forms, or whether some have more or less died out.

------
espeed
Isn't Vladimir Voevodsky's [1] Univalent Foundations [2] / Homotopy Type
Theory [3] a new candidate challenging to subsume set theory as the
foundations of mathematics [4]?

[1]
[https://en.wikipedia.org/wiki/Vladimir_Voevodsky](https://en.wikipedia.org/wiki/Vladimir_Voevodsky)

[2]
[https://en.wikipedia.org/wiki/Univalent_foundations](https://en.wikipedia.org/wiki/Univalent_foundations)

[3] [https://homotopytypetheory.org/](https://homotopytypetheory.org/)

[4]
[https://www.youtube.com/results?search_query=Vladimir+Voevod...](https://www.youtube.com/results?search_query=Vladimir+Voevodsky+univalence+foundation+of+mathematics)

~~~
kryptiskt
I wish people called these things " _a_ foundation of mathematics" rather than
" _the_ foundation of mathematics". We have been able to pick and choose ever
since foundations became a thing, and as long as they yield worthwhile results
they're worthy objects of study. We don't have to pick one and stick with it,
and it would be silly to classify stuff done with other axioms as non-
mathematics.

~~~
ellius
I don’t disagree with you, but I think the human mind is essentially
programmed to seek first causes. Part of what has made us so successful as a
species is that we want to chase patterns down to their foundations, and I
think it’s natural to feel that there is some “root” truth that explains the
entirety of existence. I think that’s why a lot of people turn to simplistic
unifying philosophies; it reduces a lot of intellectual anxiety if you have
one big idea that subsumes all others, and upon which you can fall back when
faced with uncomfortable questions. All of which is a long way of saying that
while you’re correct, I think as a species we’ll always trend towards trying
to find “THE foundation” even when the more rational thing is to “pick and
choose” like you’re suggesting.

~~~
broadwaylamb
I think the issue here is that if mathematics is talking about anything at
all, then the search for the first cause tells us what that thing is. This is
a philosophical issue separate from the sociological activity of doing
mathematics. The latter doesn't need an all-encompassing foundation to
continue to do useful and interesting mathematics, so pragmatics shrug the
issue off. On the other hand, even if mathematics is purely about "structure",
then we should still be able to organize our knowledge in a way that provides
a single foundation (even if that means showing that the different
formulations are logically equivalent).

~~~
jessaustin
One suspects that if the search for "foundations" were made rigorous enough to
analyze in any meaningful way, someone like Gödel would come along with some
sort of impossibility theorem. In other words, if we were to really examine
the idea that any particular foundation could be the best/most
natural/simplest/whatever, we'd see that it is folly.

------
eastWestMath
I’m a category theorist so I’m going to nitpick - the claim that all concrete
mathematical objects are specific sets is either tautological (a concrete
category is literally one where each object has an underlying set) or untrue
(I can think of several non-concrete categories that may be studied
“concretely” using type theory, such as the microlinear spaces of synthetic
differential geometry).

~~~
danharaj
Non-commutative spaces are definitely not sets as well. That might have
implications for foundations in the next century.

~~~
jimhefferon
I'd be interested to hear more. Any reference come to mind?

~~~
jesuslop
I'm liking
[https://arxiv.org/abs/math/0408416v1](https://arxiv.org/abs/math/0408416v1)

------
bumbledraven
Related: Metamath ([http://www.metamath.org](http://www.metamath.org)) is a
tiny formal language for writing proofs in symbolic logic, as well as a tiny
proof checker for the language. The author, Norm Megill, has used Metamath to
formalize many theorems from set theory and analysis. For example, see his
formalization of a typical proof that the square root of 2 is irrational:
[http://us.metamath.org/mpeuni/sqr2irr.html](http://us.metamath.org/mpeuni/sqr2irr.html)

------
sahilbadal
I think it is a necessary question to ask that what is mathematics which every
serious student & teacher of mathematics should confront. The title of this
book is “Foundations of Mathematics”, and there are a number of philosophical
questions about this. However, the Foundations of Mathematics should give a
precise deﬁnition of what a mathematical statement is and what a mathematical
proof is.

~~~
wastewaste
> the Foundations of Mathematics should give a precise deﬁnition of what a
> mathematical statement is and what a mathematical proof is.

I don't see a difference between precise definition and mathematical statement
in this context, so your question would be paradox. Is it wrong to ask though?
I think higher logic is supposed to deal with these types of paradoxes. Your
question, if taken rhetorical, is a statement about mathematical statements, a
metapher.

The in/-completeness theorems show that higher order logics like that are
either inconsistent or incomplete. So a theory like you are asking for, is ab
ever growing set of proofs that can't fit in a single book. Taken with a sense
of philosophy in mind, a book can only be an introduction to think further.

I was glad to read at least the intro to Univalent foundations mentioned in a
previous comment. Russels paradox defies a set of all sets in first order
logic. FOL is otherwise said to be complete, by the way, a fact often
overlooked when talking about the related liars paradox. I was pleasently
surprised to read that they just avoid the paradox by embedding theory into
theory (Universes, tbh). To me that means we can't draw elemens from an arch-
set, but have to build facts from the ground up.

