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Mathematicians Bridge Finite-Infinite Divide

quantamagazine.org

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Re: Mathematicians Bridge Finite-Infinite Divide

#11

Post by 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

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.

Re: Mathematicians Bridge Finite-Infinite Divide

#12

Post by 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.

Re: Mathematicians Bridge Finite-Infinite Divide

#14

Post by danharaj »

mrmyers wrote: post #10
danharaj wrote: post #5

Earlier quoted context omitted.

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 b…

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 disgui…

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

Re: Mathematicians Bridge Finite-Infinite Divide

#15

Post by Sniffnoy »

xamuel wrote: post #11

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 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…

A note -- if you're linking to arXiv, it's better to link to the abstract (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!

Re: Mathematicians Bridge Finite-Infinite Divide

#16

Post by colordrops »

lngnmn wrote: post #12

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…

An orthogonal issue is that most if not all humans have a hard time distinguishing which aspects (all of them?) of their reality are just abstractions.

Re: Mathematicians Bridge Finite-Infinite Divide

#17

Post by [deleted] »

lngnmn wrote: post #12

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…

[deleted]

Re: Mathematicians Bridge Finite-Infinite Divide

#18

Post by School-Cotton »

lngnmn wrote: post #12

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…

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

Re: Mathematicians Bridge Finite-Infinite Divide

#19

Post by throwawayjava »

Sniffnoy wrote: post #15
xamuel wrote: post #11

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 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…

A note -- if you're linking to arXiv, it's better to link to the abstract ( 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!

> 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...)

Re: Mathematicians Bridge Finite-Infinite Divide

#20

Post by johncolanduoni »

gfody wrote: post #4
nthcolumn wrote: post #2

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

[the divide] separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.

The article stated this in a very silly way. Infinite objects existing in nature has nothing to do with whether reasoning about certain infinite objects (e.g. The real numbers) is sound, any more than thinking about counterfactuals is impossible because they differ from the real world.
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