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The Foundations of Mathematics (2007) [pdf] (wisc.edu)
141 points by lainon on Jan 5, 2018 | hide | past | web | favorite | 34 comments



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.


So, while type theory may have the benefit of resolving the "isomorphism issue" you describe, how is type theory not also another encoding of, for example, the real numbers?


> 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.

Could you give a small example of how a simple bit of reasoning about a real (or even natural) number would work in both frameworks?


I had to chuckle when I read "as simple as a real number". If we are being strict, anything beyond the natural numbers really introduces quite a puzzle over what on earth is going on.

For example, we cannot represent the value of Pi in Hindu-Arabic notation; it stretches to some 'infinity' which is just bizarre and which we can't capture on paper. Saying it has a value opens up some interesting questions about what a number or a value actually is. I suspect people have very clear opinions which are all different from each other.


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.


NB: ICYW

  (defaxiom succ
    "The successor integer."
    []
    (==> int int))
https://github.com/latte-central/latte-integers/blob/master/...

A computer language is not just a way of getting a computer to perform operations but rather that it is a novel formal medium for expressing ideas about methodology. Thus, programs must be written for people to read, and only incidentally for machines to execute. - Preface, SICP 1st Ed. https://mitpress.mit.edu/sicp/front/node3.html


> 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 ...).


> 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.


You forgot to say that succ is injective.


Yeah, whatever, maybe some other stuff too. I should have written `epsilon N s.t. N is a minimal set that satisfies Peano's axioms`. The point is that it is not true that set theory implies some encoding. When Hilbert's epsilon is used, there is no encoding in any meaningful sense.

The main difference between type theories and set theories is that type theories have particular syntactic discipline. Everything else is or can be a feature of set theories, too.


Well, I wouldn't say type theory is superior.

Real numbers can actually be logically constructed as sets. So yes, Pi is a set.

For example, you can define the set of real numbers as the set of equivalent classes of rational cauchy sequences.

In order words, Pi is a set of rational cauchy sequences that are equivalent (this set is infinite).

In simple (non formal) words: Pi is the set of sequences of rational numbers that get closer and closer to the numerical value of Pi.

>>>> Is 3 an element of pi? If pi truly is a set

Pi being a set doesn't mean that it is a set of real numbers. In fact, that would be illogical (recursive). If Pi is a set, it must be a set of things that are not real numbers. So it is not logical to ask if 3 is an element of Pi. You can ask if 3 is equal to Pi, in which case the answer is no.


> Real numbers can actually be logically constructed as sets. So yes, Pi is a set.

You missed the point. The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. That would entail that Cauchy sequences and Dedekind cuts, for example, are the exact same thing, which they are not. Rather, they are two different ways of encoding the same abstract structure (a Dedekind-complete ordered field).

> Pi being a set doesn't mean that it is a set of real numbers.

This is dodging the question. Is the set containing the empty set a member of pi? Such a question is meaningless without fixing a particular encoding, and it has nothing to do with the intrinsic properties of a real number.


Thanks for your reply.

>The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y.

In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is - unless you assume that mathematics should aim to be a formal encoding of some pre-existing 'intrinsic reality', in which case what you're doing is physics and not math.

In physics, the debate is more open: Is the wave function literally the electron? Or is the wave function just an encoding of the electron, which is the intrinsic real thing? Or is the electron the way we humans perceive the wave function, which is the intrinsic real thing?

Anyway this is probably way too theoretical of a debate :)


> In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is

Wrong. The real numbers are literally a Dedekind-complete ordered field. The real numbers are encoded as Cauchy sequences, or as Dedekind cuts. Cauchy sequences are not Dedekind cuts: The former are equivalence classes of sequences. The latter are downward-closed sets without a greatest element. They are literally different objects.

Regarding your electron comment, the answer is straightforward: the electron is an excitation of the electron spinor field, nothing more. Fields are more fundamental than particles.


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...). 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.


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

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

[3] https://homotopytypetheory.org/

[4] https://www.youtube.com/results?search_query=Vladimir+Voevod...


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.


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.


An interesting thought brought to mind by your comments, our search for foundations (whether set theory, type theory, category theory, or higher order) reveals as much about our own minds as it does any idea of some external nature.


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).


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.


I wouldn't say it "challenges" set theory.

For this to be the case there have to demonstrable advantages over working with set theory. The main attraction of HoTT is that it allows people to do homotopy theory on computers using theorem provers. It is still a very open question whether this is an improvement over pen and paper math. It's still not clear if HoTT is just some neat technical gimmick that allows one to model homotopy theory using type theory.

This is still certainly worth investigating, but saying that HoTT "subsumes" set theory is a bit strong.

I would refer the interested reader to the discussion at https://mathematicswithoutapologies.wordpress.com/2015/05/13... (between Jacob Lurie, Shulman and Voevodsky).


It is but it’s still too new.


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).


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


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



Yeah, I would have thought that results like: https://www.scottaaronson.com/blog/?p=2725 show that set theory isn't the end all be all of mathematics.


Every formal axiom system, whether about sets or anything else, can only prove that N of Busy Beavers halt, for some (not-too-large in practice) finite number N. If you just want to maximize N, right now your best bet is set theory plus a very strong large cardinal axiom, like I0.

http://cantorsattic.info/L_of_V_lambda%2B1

Perhaps simply because it is older, set theory is way ahead of its competitors in developing a hierarchy of extremely strong (but apparently consistent) axioms to supplement its base theory ZFC.


You're confused. Only very basic arithmetic is needed to prove that any particular TM halts (if it does halt). You might have meant to say something like, "can only prove that N of the Busy Beavers are really Busy Beavers".


You're right. I should have written "halts last among its peers" or similar.


Related: Metamath (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


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.


> 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.




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