Mathematicians Bridge Finite-Infinite Divide
quantamagazine.org
quantamagazine.org
Here's the actual paper: https://arxiv.org/pdf/1601.00050.pdf
The quantamagazine article puts a lot of emphasis on the youth (34 and 27) of the researchers. I guess the journalist overlooked a more surprising age. Lu "Jiayi" Liu (whom the article briefly mentions for a preliminary result) was I think 20 years old when he made his discovery in 2012 which was a tentative step toward the stronger result in this article. I saw Liu give a talk and it was remarkable. It was in a conference where all the other speakers were very well-established logicians with many decades in the field, and then this Chinese undergrad who was probably the youngest person in the whole building... and all these well-established logicians were humbled by him.
That big grey text on the right of each arxiv paper is actually a link you can click form the pdf. Took make like N->\inf years to figure that out...
I used to complain about how heavy pdf docs are and therefore preferred websites that contained some metadata with a link to actual content, but websites these days... and at least the .pdf honestly downloads on anroid/iPhone, which are the only two platforms where I honestly care about bandwidth (and even then...)
Godels incompleteness theorems can be seen as a statement about the gap between finite and infinite mathematics. Decidability, semidecidability and undecidability can be seen as the relationship between boundedly quantified arithmetic statements, statements with one unbounded existential quantifier, and statements with one unbounded universal quantifier.
Another avenue of exploring the gap between finite and infinite mathematics is via linear logic. There the thesis is that contraction, the logical reuse of variables, is where infinity creeps into logical reasoning. Indeed logic without contraction is quite tame. Logic with unlimited contraction is wild. Surprisingly there are logics with an intermediate strength of contraction: so-called light linear logics. These can classify reasoning that embodies polynomial time computation or elementary time computation. So in another sense infinity can be measured by algorithmic complexity.
The key to decidability or undecidability is whether diagonalization is possible, not whether or not there are disguised references to infinity somewhere.
On the other hand Peano arithmetic is not only about infinite objects, and very many more than just the naturals because it is rich enough to allow you to encode other ostensibly more sophisticated infinite objects in it, it is itself an infinite object. It can't be reduced to a finitary decision procedure the way weaker arithmetics can.
Diagonalization is accounted for by my second example of conceptualizing infinity: you can't do a diagonalization argument unless you contract a variable. In particular, you can admit full unrestricted set comprehension if you can't contract to derive absurdity. Referencing section 2.3 here [1]. It was this analysis of Russell's paradox that led to the discovery of light linear logics, or so the story goes.
I'm not a mathematician, but this sounds a lot like proof by induction VS proof by anything except induction.
> RT22’s infinite structures “may make the proof easier to find,” explained Slaman, “but in the end you didn’t need them. You could give a kind of native proof — a [finitistic] proof.”
Infinity is a pure abstraction, like zero, but ill-defined (zero is an symbol for a concept of an empty slot, absence or nothing, while infinite is mere a negation of finite). Like many other concepts it might be useful, but usability does not imply existence.
> Zero is offensive and ungodly. With "five", at least there's something there to abstract. With zero, there's not even anything! How can nothing be something?
Cue angry biting of thumbs
i. Peano Axioms: (Typically) your initial element is 0 and you'll get 1 by applying the successor function to 0 and 2 by applying that to 1 and so on.
ii. Von Neumann's definition of Ordinals: {} is zero and one is {{}} and two is {{}, {{}}} and so on.
iii. Conway's definition of Surreal numbers: {|} is zero and { {|} | } is one and so on.
iv. Church encoding
and many more.
P.S. I'm not a logician and don't know whether "more of an abstraction" has any well-defined meaning and not just the meaning that might be inferred by a software developer.
For mathematical object to 'exist' only thing required is logical consistency under the rules used. Mathematical objects don't have other substance or existence than relations to other mathematical objects.
Ontological claim that abstract things that "don't exist" by some ontological definition is fine, but it does not mean much. We can't see or think anything concrete. We have never touched anything 'physical' or 'real'. It's all abstract representations in our brains.
This is some sectarian preaching in my opinion. The whole thing including these very Western philosophical sects are possible exactly because something concrete definitely exist. The DNA and the laws which make it stable and the whole life on top of it is the consequence of the fact that something concrete is out there.
This, BTW, is an millennia old debate between some branches of esoteric Eastern schools and should, perhaps, be a part of seconady school lectures.
As for definitions of infinity, the theoretical possibility of infinite series, infinite small numbers or infinitely many real numbers between 0 and 1 is nothing but mental constructs. There is not a single "concrete" infinity.
This is precisely the rule which establish existence of gods, accelerating time, higher dimensions, and other mental UFOs.
Disconnected from reality (from what is) abstract, seemingly consistent logic is what supports all the chimeras.
Read some Indian or Tibetan tantras, look at some mandalas - everything is logically consisted according to a particular sectarian logic.
Can you provide a reference to that?
> Read some Indian or Tibetan tantras, look at some mandalas - everything is logically consisted according to a particular sectarian logic.
So? "sectarian logic" isn't mathematical logic.
Also, Just to clarify: Is the specific logic wrt mathematical infinity (limits etc) also used to "establish existence of gods"?
If you mean this article, please quote the part that does that.
I've been there, done that, never made even 10 cents there! So, get to Zermelo-Fraenkel set theory, the axiom of choice, the work of Kurt Gödel and Paul Cohen (I still have the copy of Cohen's paper Max Zorn gave me!), etc. A friend worked in forcing arguments, Ramsey theory, etc. and never made even 10 cents there either.
I climbed out of that dark basement and don't want to go back!
Maybe because I didn't start out with money making in mind (not to say I don't want to make money. I do. I do.)
Also how long did the whole process take for you? ... If you make up your mind in advance, as I have, that you're going to be in there for 10 years give or take, (in 8-hour days; so 20 calendar years if you spend 4 hours a day), and try to make a living on the side, does it still feel as dark?
The most intense time was in a course in axiomatic set theory in an NSF summer math program at Vanderbilt.
The next fall, I was in a course in real analysis, and the prof started with foundations. After the first test he wanted to "see me". The exercises on the test were all trivial except one, and I got it only in the last minute or so so wrote quickly. I used little omega for the ordinal of the natural numbers without defining it. I told him that from the course I'd taken the previous summer I thought that that was standard notation. Apparently he didn't know that. After my explanation, he saw that my solution was correct and one step shorter than his. Then I asked him what he wanted to "see me" about, and he said "Now, nothing.". Gads. I got my Ph.D. later from a better program at another university.
Occasionally later I touched on that material.
That was all.
No way did I spend or want to spend 10 years in that basement. And no way did I want to do original research with that material.
Some of the pure math I studied I liked and still like a lot, and some of it is a crucial pillar of the crucial applied math core of my startup.
Besides, "pure" and "applied" distinction is hilariously subjective and stupid. IME a lot of theoretical CS is more-or-less accurately characterized "pure math" and also far more financially remunerative than the bullshitty pointless PDE hacking lots of "applied math" folks do. Pure vs. applied is a stupid perspective these days because it almost exclusively defines a delineation among communities of mathematicians during the late 20th century; "useful in the next 3 years to ad tech folks" vs. "long-term/foundational importance in science and engineering" is a more useful and relevant distinction.
But once people choose to focus on pure mathematics, don't shit on them for making that choice. They KNOW it's not a great financial commitment. It's like telling a committed humanist that "no one pays for poetry". Like, they get it already... (and besides, sometimes you're wrong and then you're that asshole that always doubted them.)
Plus, plenty of pure mathematicians make high five/low six figures working 12 months with free trips all over the world. Or 9 months without the trips but with 3 months of freetime every year. And in both cases with near perfect job security. They don't have any social currency in the startup $$$$$$ obsessed world, but I bet they spend a fuckload more time with their kids and enjoy a lot more sunsets than any of startup founders. And probably make more money than the 90% of "failure cases" in the startup world too...
This article is great case-in-point. All the profiled folks are sure to live very comfortable lives doing work they love patronized by lovers of the mathematical arts (or anyways live off of rich endowments one way or the other) without ever having to figure out how to nickle and dime customers.
And none of this is to say that building businesses isn't valuable, since that's more-or-less taken for granted by the current venue. But it takes all types.
I was saying that I, personally, find foundations as in the OP down in the basement, dark and too far from applications in any sense.
For making money as a full prof of math, first have to get there, and that usually takes over 10 years if make it at all. Yes, it's possible to play the academic game; heck one paper in math I published is pretty, surprising, etc. but I can't imagine that it will ever be useful directly or even indirectly even by several steps of indirection for the foreseeable future if ever. I've seen people publish such things and make some progress in an academic career; to me that's playing an academic game; I chose not to do that.
When I was a prof (I didn't want to be but did it for a while trying to help my wife in her illness), it seemed to me that I was getting paid by students, pizza parlor owners and workers, auto dealership owners and workers, farmers, etc., and for them I wanted my work and teaching to be useful -- to me, personal curiosity, art, etc. didn't count. I really wanted the department to be clinical, professional, practical, like law and medicine, i.e., welcome people from outside academics with real problems and then seek to solve those problems. When can't solve a problem, then maybe that will be a good research direction; if make good progress, then already have one application!
For applied math, determine that not by PDEs but by what can find that is useful. Early in my career around DC for mostly US national security, I found lots of such applied math. And my startup, while more focused, is basically some applied math. PDEs? I had very little to do with those; the one case was the Navier-Stokes equations, and they were so difficult to work with that the project wasn't making much progress and I was pleased to move on to other topics.
It's not easy to see what pure math is the more useful. For the pure math I do respect, especially for utility, it appears that in some vague, large sense the results are fundamental, important broadly, and, eventually, inescapably relevant, but that is a difficult judgment call. Functional analysis? Sure. Algebraic geometry? Less sure. Foundations? Slim chance.
I was not trying to give career advice to other people possibly interested in math but just commenting that I found the foundations as deep as in the OP just too far down in a dark basement. In math, that's an old remark about foundations -- "Drop what you are doing trying to be useful and come with me down into the dark basement and wrestle with really subtle issues down there." Standard, old remark in math.
You read lots of stuff between the lines I wrote, stuff not really there.
Let's take an example from TCS: Quantum computers. Initially, they were only a theoretical exploration of the the additional power (if any) that Quantum mechanics can give in computing things. Then Shor came around and shook up our understanding of the world.
Many such examples abound in the world of mathematics. And lastly, even if a subfield provides no use to "common" humans, how should that matter? People should be free to study what they want.
> Who gets to decide what is and isn't useful?
There is a recipe for rabbit stew that starts out "First catch a rabbit". Well my own recipe for applied math starts out "First find an application."
I don't define applied math as just mathematical physics or its connections with, say, mechanical or electrical engineering.
For me, applied math is math that has been applied. Sure, you mention number theory and cryptography; so, now at least in part they are applied math. There is also finite fields and error correcting codes -- I took a whole grad course on coding theory. I had a good background in abstract algebra and was torqued that the prof was sloppy with the math. But coding theory is important, and for me that makes abstract algebra applied math. Just how R. Hamming saw to use finite field theory I don't know -- curious insight. My undergraduate honors paper was on group representation theory, and that is applied math at least for its connections with molecular spectroscopy. In fact, my work was stimulated by some people from the chemistry department that came to the math department for help with group representation theory. IIRC, the quantum mechanics part was in part from E. Wigner.
> And lastly, even if a subfield provides no use to "common" humans, how should that matter?
If the work is being paid for, then commonly it matters a LOT to some of the people paying for it.
> People should be free to study what they want.
Of course. For some years, I studied violin. I made some progress but set violin aside to have more time for my startup. Ah, but, no one paid me for studying violiin!
People misread my post: First, the only math I was criticizing was foundations as deep as in the OP. Second, my criticism was only for my own personal values and opinion. Sure, maybe work in Ramsey theory will be as important in applications as the Riemann integral -- I doubt it, but maybe.
I spent some months in that dark basement; others are welcome to do that if they wish; I wish I hadn't and wouldn't do it again.
Taking an example from TCS: Probabilistically checkable proofs. They allow you to encode a proof for a statement so that you checking the proof requires looking at in only a few locations. Back when they were conceived, in the early 90s, all constructions were efficient only asymptotically, with galactic constants. Only in the past 2 years have constructions of PCPs been realized that are sufficiently efficient for (some) applications to checking program executions.
PCPs were for years what most theorists would consider a core theoretical object with not many hopes of finding application, and indeed 99% of research in the field focused on "negative" uses of PCPs for finding hardness of approximation results.
> If the work is being paid for, then commonly it matters a LOT to some of the people paying for it.
What makes an industry with a lot of money in it, like online advertising, worth more than study of something completely theoretical?
Not "disingenuous" at all: I want to apply math, especially to making money. To make money is the main reason I studied math; I wanted math to help my career, to make money, to support a family. If not math, then maybe physics, some part of engineering, etc.
> what is and isn't applied can change rapidly in the course of a few years.
My experience is that that doesn't happen very often. But again, once again, over again, yet again, I am not, Not, NOT, N.O.T. -- clear enough -- running down pure math. NOT doing that. I just wrote that foundations as in the OP was too far from applications for me. I have been totally, overwhelmingly, crystal clear about this point in this thread, and there was nothing in my first post that ran down pure math. Again, yet again, ..., I was talking about my view for me of foundations such as in the OP. You misread my first post and didn't read my other posts.
If someone found some applications for PCPs, then good for them, and that makes PCPs applied math.
Better techniques for proofs of correctness obviously would be good work in computer science with plenty of valuable applications.
> What makes an industry with a lot of money in it, like online advertising, worth more than study of something completely theoretical?
Nothing, but most sources of financial investment want a financial return. That's much of how our economy works.
To be blunt and frank, Congress votes money for math mostly for US national security and because of the role of math in WWII and off and on since then. If some people at the NIH tells Congress that they like math, too, then that will help. But the remark at the beginning of the movie on Nash is basically correct: "Mathematics won WWII". Congress can be slow on the uptake, but they tend to believe that math did win WWII and had a big role in GPS, Keyhole, stealth, design of the core of fission and fusion bombs, etc. Congress takes US national security quite seriously. The other biggie for Congress is the NIH -- if only because most people in Congress have gray hair and want research that will help them if they get sick.
It remains, if you want a job in math, especially connected with computing, then those jobs are still where I started my career (and should have stayed there) within 100 miles of the Washington Monument.
You speak like someone who knows the price of everything and the value of nothing.
A lot of "pursuits have value outside of money"; e.g., for some years I pursued violin.
But if a person is being paid to do research in mathematics, then usually in some sense commonly the people paying will want to know if the work is or will soon become useful. In fact, there was the David Report that severely criticized Federally funded math research that seemed to have no intended connection with applications. The theme of that criticism was if the math is being pursued just as art, then fund it like art.
I was giving my personal opinion and not trying to change the opinion of anyone else. Read my other responses here.