Mathematicians Bridge Finite-Infinite Divide
quantamagazine.org
Mathematicians Bridge Finite-Infinite Divide
1–10 of 47 posts
Re: Mathematicians Bridge Finite-Infinite Divide
#2Where is this divide? It is some concept I am unaware of?
Re: Mathematicians Bridge Finite-Infinite Divide
#3Ah, 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!
Re: Mathematicians Bridge Finite-Infinite Divide
#4Where 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.
Re: Mathematicians Bridge Finite-Infinite Divide
#5Where is this divide? It is some concept I am unaware of?
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.
Re: Mathematicians Bridge Finite-Infinite Divide
#6Ah, 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 pap…
I'm on my way down but haven't seen the darkness yet :)
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?
Re: Mathematicians Bridge Finite-Infinite Divide
#7Where is this divide? It is some concept I am unaware of?
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…
You just gave me a new frame for thinking about a few things I'd already learned, as well as some interesting leads on questions I didn't even know to ask. Thank you!
Re: Mathematicians Bridge Finite-Infinite Divide
#8> 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.
Re: Mathematicians Bridge Finite-Infinite Divide
#9Ah, 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 pap…
I'm on my way down but haven't seen the darkness yet :) 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 livin…
One way and another, starting in my senior year in college, I spent a year or so in that dark basement.
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.
Re: Mathematicians Bridge Finite-Infinite Divide
#10Where is this divide? It is some concept I am unaware of?
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 disguised references to infinity somewhere.