On June 20th, José Ferreirós (University of Seville) will give a talk at the Centre Gilles-Gaston Granger (Aix-en-Provence), titled:

“Dedekind and the Axiomatic Method”

https://www.cggg.fr/agenda/seminaire-avec-jose-ferreiros

🕤 9:30–11:30 (CEST)
📍 Maison de la Recherche, Room 2.44

This session will include a discussion on Dedekind’s role in the emergence of the axiomatic approach, with further reflections on mathematical practice.

(Feel free to contact paola.cantu@univ-amu.fr to confirm attendance.)

#philosophy #mathematics #philosophyOfMathematics #seminar #Dedekind #cggg

“How The Square Root Of 2 Became A Number”, Quanta (https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/).

On HN: https://news.ycombinator.com/item?id=40750230

#Numbers #IrrationalNumbers #Mathematics #Maths #PythagorasTheorem #Dedekind #Cantor #History #Reals #Irrationals

Arg, my memory is a sieve.

#Dedekind expert on Mastodon, where are you? I can't remember your username, your real name, nor find comments here by you, nor your website 😠

[EDIT: found her!]

When you put off your doctoral dissertation to work on a stubborn problem.

#maths #math #dedekind

https://www.quantamagazine.org/ninth-dedekind-number-found-by-two-independent-groups-20230801/

Big news in the world of math because I know that you care. 😉

"Mathematicians Discover The Ninth Dedekind Number, After 32 Years of Searching

Undeterred after three decades of looking, and with some assistance from a supercomputer, mathematicians have finally discovered a new example of a special integer called a Dedekind number.

Only the ninth of its kind, or D(9), it is calculated to equal 286 386 577 668 298 411 128 469 151 667 598 498 812 366, if you're updating your own records. This 42 digit monster follows the 23-digit D(8) discovered in 1991."

🔗: https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Math #science

"Van Hirtum believes a similar jump in computer processing power will be required to calculate the 10th #Dedekind number. 'If we were doing it now, it would require processing power equal to the total power output of the sun,' he said, which makes it 'practically impossible' to calculate."

Just hold my beer while I plug into my Dyson sphere.

https://www.livescience.com/physics-mathematics/mathematics/mathematicians-finally-identify-seemingly-impossible-number-after-32-years-thanks-to-supercomputers

Mathematicians Christian Jäkel and Lennart Van Hirtum et al. simultaneously discover the 42-digit Dedekind number after 32 years of trying.

The exact values of the Dedekind numbers are known for \(0\leq n\leq9\):
\(2,3,6,20,168,7581,7828354,2414682040998,\)
\(56130437228687557907788,\)
\(286386577668298411128469151667598498812366\)
(sequence A000372 in the OEIS)

🔗 https://scitechdaily.com/elusive-ninth-dedekind-number-discovered-unlocking-a-decades-old-mystery-of-mathematics/?expand_article=1

🔗 https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

Summation formula👇
Kisielewicz (1988) rewrote the logical definition of antichains into the following arithmetic formula for the Dedekind numbers:
\[\displaystyle M(n)=\sum_{k=1}^{2^{2^n}} \prod_{j=1}^{2^n-1} \prod_{i=0}^{j-1} \left(1-b_i^k b_j^k\prod_{m=0}^{\log_2 i} (1-b_m^i+b_m^i b_m^j)\right)\]

where \(b_i^k\) is the \(i\)th bit of the number \(k\), which can be written using the floor function as
\[\displaystyle b_i^k=\left\lfloor\frac{k}{2^i}\right\rfloor - 2\left\lfloor\frac{k}{2^{i+1}}\right\rfloor.\]

However, this formula is not helpful for computing the values of \(M(n)\) for large \(n\) due to the large number of terms in the summation.

Asymptotics:
The logarithm of the Dedekind numbers can be estimated accurately via the bounds
\[\displaystyle{n\choose \lfloor n/2\rfloor}\le \log_2 M(n)\le {n\choose \lfloor n/2\rfloor}\left(1+O\left(\frac{\log n}{n}\right)\right).\]

Here the left inequality counts the number of antichains in which each set has exactly \(\lfloor n/2\rfloor\) elements, and the right inequality was proven by Kleitman & Markowsky (1975).

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Sequence #Discovery #Mathematicians #Challenging #RichardDedekind #DifficultProblem #MathHistory #Pustam #ChallengingProblem #EGR #PustamRaut