This Man Is About to Blow Up Mathematics

Basically after Godel's theorems people thought that for the most part they weren't an issue and it was completely fine to work inside the usual axioms of mathematics, never requiring anything more like say, the axiom of large cardinals, what some people have tried doing in the past couple of decades to confirm this sort of idea is called reverse mathematics, tracing core theorems back to find out what axioms are necessary to prove them. It turns out the ZFC (usual axioms) is more than enough to prove the vast majority of mathematics, with the only statements unprovable in ZFC being set theoretic and for very odd systems (it actually turns out that you could have a FAR weaker set of axioms and still have most of mathematics work out just as well). What Friedman did though is show that there were statements about the rationals that required axioms outside of ZFC to prove them, this means that idea that most mathematicians can work with just ZFC and nothing else may be false, since there could be a variety of statements relevant to their work that they won't be able to prove just using ZFC.

This is great. Can you expand upon what these statements about the rationals are? How do about this area of mathematics? It sounds very interesting.

How do you know about this area of mathematics? It sounds very interesting.*

>child prodigies never amount to anything

There you go faggots. This guy was a child prodigy. Ph.D from MIT at 17. Assistant Professor at Stanford at 18.

He's got a Wikipedia page. That makes him part of the top 0.1% of professors.

His brother is also a mathematician and also a child prodigy, receiving his Ph.D from MIT at 23.

>inb4 a mountain of rationalisations

If you have some time what you may want to do is check out Friedman's videos on logic for undergrads youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ/videos
leading up to a video explaining his results
youtube.com/watch?v=JG9_kygjgc0
all of them are pretty basic (I say this because I don't know your background in logic/math so it'd be a bit difficult to properly explain what he did, if anything it's better to hear it from the horse's mouth). I personally learned about basic logic from various texts, but these videos seem to go through enough to understand his work.

>cs.nyu.edu/pipermail/fom/2017-July/020533.html
crackpot

You provide no evidence. Go away

> reverse mathematics
Legitimately surprised, I have been attempting something similar w/r/t other subjects. Eg:
> what axioms are necessary in order to prove XYZ theorem in psychology, or philosophy, or physics etc

WHY ISN'T ANYONE STOPPING HIM! Do you know how many times I rely on math in my day to day life? What are we gonna do if he blows up math? It'll be like returning to the stone age I tell you!

Then you'll like what Friedman is doing, he basically wants to do exactly that, find a way of looking at every field and finding out what are the necessary axioms, tried doing that with music and got decently far, though this sort of idea of tracing things back to their foundation is only a major topic in mathematics, I have yet to see other fields attempt this (barring physics). This is likely due to the fact that many fields don't prefer a reductionist view point or are unable to properly work things through, though even with emergent phenomenon there is usually some underlying cause that can then be worked into the larger framework, so I think taking potshots at reductionist methods for that reason is a bit faulty

Re-reading this, am I correct in understanding that ZFC is going to need to be expanded in order to keep using/believing in rationals?
Neat, I'll keep my eye on him. Let me know if I'm doing it right, I was trying to think through what would be the axioms of something like physics, and I was astonished at how many assumptions you need to make in order to 'do' physics.
> There is a mind independent world
> This mind independent world can be investigated
> History has happened
> Probably something else
Is this similar to what he is doing?

Thanks for the link, I'll check it out. I took a set theory course a while back ago and a few logic courses. I could probably understand the basics.

I am and My understanding is they are going from theorems to axioms. Hence the name "reverse math". They've shown that a basic set of axioms in mathematics is capabale of proving most of what one studies as an undergraduate in mathematics. I supppse th assumption is ZFC is capable of proving most theorems mathematicians care about on a day to day basis. Friedman apparently is showing theorems that are outside of ZFC. I need to watch the video linked and read some papers to say any more

Hang on, I thought we already knew that there are theorems that exist 'outside' of ZFC, and that the 'big thing' is that Friedman is showing that ZFC is not sufficient to prove theorems which mathematicians had previously thought could be proven via ZFC, and that in order to prove those theorems, one must 'take on board' additional axioms?

If I'm understanding correctly, this could mean that ZFC is going to lose its status as the 'default' set of axioms which mathematicians work with, and that proving some stubborn theories like the Riemann function may be possible within this 'new' set of axioms.
I could be wrong though

I don't know enough to say anything intelligent on the subject. Seems like Knows.

Unless you are him, in that case, maybe you can clarify more as I may have mis understood you?

We can still use the rationals just fine in ZFC, the main point is that are seemingly obvious/intuitive properties of the rationals that exist beyond ZFC and this might apply more broadly to mathematics as a whole. There is a difference between the philosophical underpinnings of science (this is under the purview of the philosophy of science, a commonly known developer being popper) what this project and Friedman are looking at concern more the principles from which we derive various laws and theorems, an example of an "axiom" in physics to this effect is the equivalence principle which underlies all of GR, or the axioms of QM found in nearly every QM text, the point of reverse mathematics is to understand the interaction between what we assume and the larger scope of the things based on these assumptions, fruitful since it can then guide you to unexplored territory more easily yet still in a tractable manner than can be connected to the original subject, in this case a possible number theoretic or combinatorical property of the rationals not know which may be of great interest to the community, though there is still so much more to be done in reverse math and unraveling it's full applications.
No problem
To clarify, we already knew that there exist theorem's unprovable (outside) in ZFC, this was Godel's magnum opus, however mathematicians figured that they wouldn't run into these issues since most of mathematics doesn't use nearly the full power of ZFC and as such shouldn't run into this problem. Friedman showed this assumption doesn't hold as there are properties of the rationals (which are at a low level in the set theoretic hierarchy and as such were thought to have any and all of it's properties provable in ZFC ) that aren't provable in ZFC.

I am I have given a response here

Thanks for the clarification. This is from

Feeling like a brainlet, let me see if I understand. We had/have intuitions about the rationals. (PS what are these intuitions)
We thought we could prove these intuitions using ZFC
This is not the case?
Can you give an example of how this reverse mathematics would work in practice?

Kind of, there are properties of the rationals (probably the most basic and widely used object in all of mathematics) that is not provable within ZFC. If the most basic object we worked with has properties outside of our usual axioms, is it the case that there are interesting properties throughout all of the mathematics that we have missed simply because we have restricted ourselves by working in ZFC? Unfortunately there aren't too many examples of reverse math producing these new interesting properties of usual objects, nor the production of new objects either, this partly due to the fact that the program is relatively young (not even 40 years old) and remains a niche subject. There's also the issue that it's still very much in development as no one really knows the ramifications of adding a new axiom or even tweaking the usual. A concrete example however was stated in the article and explained in the video I posted.

will this guy surpass Mochizuki?

This is a field of mathematics known as recursion theory, or computability theory. Most CS/math majors think of CS when they hear these terms. No, this is a sub-field of mathematics, researched by mathematicians.

Stephen Simpson is another major mathematician in this area.

Computability theory represents the larger field of which this program is a subfield, yes, (I am also well aware that entail very heavy mathematics), I was simply pointing out this goals of this specific program, as computability theory as a whole has many different topics of interest each of which doesn't necessarily overlap in the easiest to explain manner, also reverse math is the one most relevant to the article and one of the easier disciplines to explain.

So Friedman is giving new insights into what we thought were well understood mathematical objects?

Agreed with what you said. I find it a bit taxing trying to explain to others that computability theory is a field of mathematics. Most are exposed to the basics of it in an automata theory course and draw incorrect conclusions it's a dead field, or not a very deep one, or that it is something computer scientist exclusively study.

In a sense, yes.
I never really understood why people just assume that anything having to do with topics that might be relevant to CS are automatically CS, in mind at least I've always considered theoretical CS to be a field of mathematics in it's own right, in a similar fashion just cause computatbility theory of great importance to CS it shouldn't be considered any less of a mathematical field itself, it very much hits at the core of what we do in math, the biggest example I know of being the negative solution of one of hilbert's problems.

Indeed. Anyone studying logic in the mathematics department has to go through the same set of requirements to obtain a PhD in Mathematics as any other student. I chalk it up to ignorance, which is probably partly related to the fact only basic logic is covered in most intro to proof courses. After that, there doesn't seem to be much exposure to the subject for most undergrads. I was fortunate enough to take a course in recursion theory as a math major. If curious, we studied from the 3rd edition of Boolos' book on Computability.

Lost

Thread actually got better with time

Agreed.

>outlier
wooooow

Why did you have to pick such a gay website? I really don't even want to read the article now.

same guy

>Gödel had effectively shown that every axiomatic system, no matter how comprehensive, is vulnerable to irreparable holes. Filling those holes by creating a stronger system would only yield new statements that cannot be proven—so that an even stronger system would be needed, and so on, ad infinitum.

Is this true? Wikipedia disagrees. See below

>For any such formal system, there will always be statements about the natural numbers >that are true, but that are unprovable within the system. The second incompleteness >theorem, an extension of the first, shows that the system cannot demonstrate its own >consistency.

Is she ASHKENAZI? I probably don't even need to check.

BIG

(((friedman)))

Don't read the article. Read Friedman's actual work. Friedman stated he had no control over what the brainlets wrote. You're a brainlet yourself for not figuring that out. Friedman has his stuff on FOM and other anons explained what he's working on. You'd be too much of a brainlet to understand this, because you never bothered even reading the thread. Leave Veeky Forums and don't return. Veeky Forums will directly improve on your departure.

user who wrote most of the answers here, just read the thread, most of it is explained

no u

>And it consumes him still as a 68-year-old retired math professor living on a leafy street in suburban Columbus, Ohio, sleeping for a few hours at a time, twice a day, so as to free up time to think.

Whew I thought it would take longer for me to find someone who wakes up to decide if something is true or false as well. Sleep is a waste of time.

If you're interested in things such as proof assistants and univalent foundations/axioms of mathematics that can be used to generate proofs from solutions then start with Homotopy Type Theory cs.cmu.edu/~rwh/courses/hott/ (lectures included) the book is also free here homotopytypetheory.org/book/

The computational content of Voevodsky’s Univalence Axiom in Homotopy Type Theory remains unclear unless you want a billion page proof so after using HoTT to formally define a lot of things they expanded abstractly into the Bezem-Coquand-Huber model of cubical sets carried out in constructive set theory and now research is ongoing to create a working syntax for higher cubicle type theory which gives more abstract axiomatic freedom meaning automated proof generation programs in the future that can actually be read and understood by humans.

ASHKENAZI

>blow up
>math
SORRY KID
it's already been done