
Mathematicians Bridge Finite-Infinite Divide - lolptdr
https://www.quantamagazine.org/mathematicians-bridge-finite-infinite-divide-20160524
======
xamuel
A genuine breakthrough in the narrow realm of reverse mathematics. (The
article exagerates that into a breakthrough _in general_ , par for the course
for science journalism.)

Here's the actual paper:
[https://arxiv.org/pdf/1601.00050.pdf](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.

~~~
Sniffnoy
A note -- if you're linking to arXiv, it's better to link to the abstract
([https://arxiv.org/abs/1601.00050](https://arxiv.org/abs/1601.00050)) rather
than directly to the PDF. From the abstract, one can easily click through to
the PDF; not so the reverse. And the abstract allows one to do things like see
different versions of the paper, search for other things by the same authors,
etc. Thank you!

~~~
throwawayjava
_> not so the reverse_

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...)

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robinhouston
The previous discussion of this article was interesting too:
[https://news.ycombinator.com/item?id=11763080](https://news.ycombinator.com/item?id=11763080)

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gweinberg
I took a course in combinatorics as an undergrad, so I know what Ramsey's
theorem is. But our professor didn't go over the proof of Ramsey's theorem,
our professor said he couldn't expect us to understand the proof because he
didn't understand it himself. I might have gotten more out the article if
there was some explanation as to what a "finitistic" proof is, and how it
differs from the "infinitistc" proof. Maybe the concept is too complicated to
explain in a short article.

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nthcolumn
Where is this divide? It is some concept I am unaware of?

~~~
danharaj
In principle a statement that involves only finite sets of natural numbers can
be proved by exhaustive calculation. A statement involving infinity in an
essential way cannot. This means there is a gap in certitude between finite
and infinite mathematics. There's many ways to describe this divide. For
example, an arithmetic statement involving unbounded quantifiers. You can
measure how much such a statement fails to be finite by counting the
alternation of universal and existential unbounded quantifiers. This is a
measurement of the naive logical complexity of a statement.

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.

~~~
mrmyers
First-order arithmetic with bounded quantification is decidable, but so is
arithmetic with unbounded quantification but no multiplication (just
addition). So is the elementary theory of real numbers, and elementary
geometry. Meanwhile, there are plenty of small, finitary theories that are
undecidable.

The key to decidability or undecidability is whether diagonalization is
possible, not whether or not there are disguised references to infinity
somewhere.

~~~
danharaj
I think the distinction here is that even though a theory like Presburger
arithmetic is about the infinite set of natural numbers and similarly for
Euclidean geometry that they are still finitary objects precisely because they
are decidable: the entire theory can be reduced to a finite object, the
decision procedure.

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.

[1] [http://www.brics.dk/LS/96/6/BRICS-
LS-96-6.pdf](http://www.brics.dk/LS/96/6/BRICS-LS-96-6.pdf)

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snissn
Good.

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hzhou321
The paper is too abstract for me but it seems to mean that we don't really
need the concept of infinity (for theorems in finite domain). I like that.

------
vanderZwan
> _The colorable, divisible infinite sets in RT22 are abstractions that have
> no analogue in the real world. And yet, Yokoyama and Patey’s proof shows
> that mathematicians are free to use this infinite apparatus to prove
> statements in finitistic mathematics — including the rules of numbers and
> arithmetic, which arguably underlie all the math that is required in science
> — without fear that the resulting theorems rest upon the logically shaky
> notion of infinity. That’s because all the finitistic consequences of RT22
> are “true” with or without infinity; they are guaranteed to be provable in
> some other, purely finitistic way. 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.”_

I'm not a mathematician, but this sounds a lot like proof by induction VS
proof by anything except induction.

~~~
vanderZwan
Maybe I should have clarified. I recall being taught induction in my first
year of physics, but the maths professor told us that it is generally frowned
upon as a proof because it does not give any insight. So if there is a choice
between two proofs, the other option is almost always more useful. This last
sentence sounds a lot like that:

> _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.”_

------
lngnmn
There is no such thing as Infinite outside your heads. This is actually a
pattern - a false dichotomy with a pure abstraction produced as an abstract
opposite or an abstract result of negation of some other concept or a named
entity. Applied Hegelian nonsense if you wish.

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.

~~~
umanwizard
How is "zero" more of an abstraction than "five" ?

~~~
gizmo686
Zero was invented long after the rest of the natural numbers (to my knowledge
this is the case in every culture that independently invented zero). At least
in the case of the Greeks, its status was somewhat controversial; I am not
sure how other societies viewed it once it was invented.

~~~
johncolanduoni
If we find out some alien race started using positive numbers and zero at the
same time, do they then become the same level of abstraction?

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graycat
Ah, foundations of math, start with applied math for making money, descend to
applied math that doesn't make money, descend to pure math, descend to
foundations, and, there, down in the dark basement try to make some sense.

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!

~~~
omginternets
Intellectual pursuits have value outside of money, you know.

You speak like someone who knows the price of everything and the value of
nothing.

~~~
graycat
You seriously misread my post.

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.

