61–70 of 107 posts
Does mathematics contain a lot of theorems that rely on unproven assumptions?
All knowledge ultimately relies on some self-evident first principals which are not demonstrable.
Technically, we can derive one fact — there is something rather than nothing:
Asking the question is itself proof.
It is so amazing that I can read articles like this for free
I can't get enough of these mathematical stories proving/disproving conjectures. I think they show a more human part of mathematics, which I rarely got to see in my college courses.
It is so amazing that I can read articles like this for free
I feel the same gratefulness. While I was reading I was thinking that whoever wrote this did a great job to make it understandable. Without their input I believe I would have never been told this story. I can't get enough of these mathematical stories proving/disproving conjectures. I think they show a more human part of mathematics, which I rarely got to see in my college courses.
Does mathematics contain a lot of theorems that rely on unproven assumptions?
Godel's incompleteness theorems say that all mathematics that is complicated enough to encode basic arithmetic must rely on unproven assumptions. Unproven assumptions are the basis of all mathematics. For example, even numbers rely on very unproven assumptions. We assume, for example, that there is always a number following another number, but it is not at all obvious what that means.
Assuming you're referring to natural numbers or integers, that's not an assumption:
tl;dr: the conjecture is that if you start with three touching circles, each of which has an integer curvature (i.e. 1/r is a whole number) and you then draw a circumscribing circle around those, then starting filling in the gaps between the circles with ever smaller circles, every one of those smaller circle will also have an integer curvature. Turns out, they don't, but no one actually sat down to do the grunt work…
That the curvatures are integers is proved (noticed by Soddy in 1936 as "a fairly straightforward consequence of Descartes’ equation", as the article mentions), and a lot more is also proved about the integers' distribution; the conjecture specifically is about whether every sufficiently large “admissible (passing local obstructions) integer is the curvature of some circle in the gasket” — see the second page of http…
Earlier quoted context omitted.
Unless it's not. I'm sure lots of mathematicians have open bets about that. I still remember my supervisor hoping for no Higgs "because then physics will be boring for the foreseeable future".
I think anyone that's betting on it being false is nuts at this point. It would be astounding if it held for the first three trillion then broke down. Generally patterns like this get more regular as the numbers get bigger not less.
Earlier quoted context omitted.
Counterpoint: am American and I read it exactly as intentioned the first time. It didn't even occur to me to read it in the context you did until you mentioned it. I don't watch network news though.
I don’t watch network news either and had the same reaction as OP reading “two students shoot down” A better headline could be “Widely Believed Math Conjecture shot down by two students.”
Background:
- Apollonian circle packings is the study of how circles can fit into a larger circle.
- Rather than using diameter to measure these circles, mathematicians employ curvature — the inverse of the radius. The smaller the circle, the larger its curvature.
- When the first four circles have an integer curvature, all subsequent circles in the packing will also have integer curvatures.
- Mathematicians later focused on identifying which integers emerge as the circles shrink and the curvatures grow.
Key Developments:
1. Local-Global Conjecture: Elena Fuchs proved in 2010 that curvatures conform to a certain relationship. This led to the belief known as the local-global conjecture, which claims that all possible numbers within each category must appear in the circle packings.
2. Testing the Conjecture: James Rickards created software to examine any desired arrangement of circle packings. When researchers Summer Haag and Clyde Kertzer started using the software, they anticipated observing the regular patterns of the local-global rule.
3. A Surprise Discovery: After conducting extensive plotting, Haag observed patterns that didn't align with the local-global conjecture. This suggested that the conjecture may not hold universally.
4. Disproving the Conjecture: Upon further analysis, it was determined that the observed patterns indicated that the local-global conjecture was false. The team developed a rigorous proof, utilizing the principle of quadratic reciprocity, which explained why certain curvatures can't be tangent to each other.
Implications:
- The discovery was met with significant interest and surprise in the mathematical community.
- The work questions the validity of other conjectures in number theory that have been largely assumed to be true.
Conclusion:
The study of Apollonian circle packings led to the challenge and ultimate disproval of the previously accepted local-global conjecture. This outcome underscores the importance of testing long-held beliefs in mathematics and the potential surprises that can emerge from seemingly simple problems.
Earlier quoted context omitted.
All knowledge ultimately relies on some self-evident first principals which are not demonstrable.
ALL knowledge does? Can you prove that? :)