Fun from Joel David Hamkins https://www.youtube.com/watch?v=DGIA2U2iPCk #mathhistory #mathsong

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

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

🎉The third (and final!) paper from my Ph.D. dissertation is now out at @PhilSciJournal as 👀 First view article 👀 #philsci #histofscience #mathhistory
🧐Interested in how mathematical communities react to new ideas?
Read on:
(if you don't have 🔐, please pm me!) https://www.cambridge.org/core/journals/philosophy-of-science/article/abs/connecting-the-revolutionary-with-the-conventional-rethinking-the-differences-between-the-works-of-brouwer-heyting-and-weyl/9376F606893AEDB9AD887548C59D6099

@rrogers 😯
Do you think that's still at play, though? After all, the UK has had mathematical excellence at the top of its university system for centuries now. I had the privilege of studying at #Trinity in #Cambridge, where you trip over mathematical history constantly. (Great opportunity for learning some history of maths for me!)
#maths #education #MathHistory

von Neumann’s letter has it all: a discussion of the theory of spectra of Hermitian operators, inquiries about spin-geometries and not least of all, gossip. Lots of gossips. 🔗 https://www.cantorsparadise.com/john-von-neumanns-1935-letter-to-oswald-veblen-3acbe1b69098

#vonNeumann #JoohnvonNeumann #CantorsParadise #Hermitian #HermitianOperator #SelectedLetters #HermitianOperators #Neumann #MathHistory #Mathematics #MathsHistory

von Neumann’s letter has it all: a discussion of the theory of spectra of Hermitian operators, inquiries about spin-geometries and not least of all, gossip. Lots of gossips. 🔗 https://www.cantorsparadise.com/john-von-neumanns-1935-letter-to-oswald-veblen-3acbe1b69098
#vonNeumann #JoohnvonNeumann #CantorsParadise #Hermitian #HermitianOperator #SelectedLetters #HermitianOperators #Neumann #MathHistory #Mathematics #MathsHistory

Random physics history fact:
Paul Dirac's favorite drink was water.

#Physics #History #Science #STEM #Dirac #Math #MathHistory #physicshistory #randomthoughts #nerdlife #Nerd #quantum

#introduction So I have been told that I should introduce myself in this platform, and what a better way to do that then to share a new publication!
Excited to share that my paper “From philosophical traditions to scientific developments: reconsidering the response to Brouwer’s intuitionism” is now available in Synthese: https://doi.org/10.1007/s11229-022-03908-3 #philsci #MathHistory Interested in reading and don’t have access? PM me for a copy 📄

#Math
#MathHistory
#STEM
#MathEducation

December 1st, 1792.
Nikolay Ivanovich Lobachevsky is born.

And Tom Lehrer sings about it!

https://youtu.be/_lf1mB8OZeg

The #SuryaSiddhanta's methods for calculating the positions of the sun and moon at times specify that the absolute value of the jya of an arc must be added to some value in some cases, and subtracted in others.

Since the jya is the predecessor to the modern sine, in this we see the earliest inklings of negative values of trigonometric functions. It's interesting how ideas develop.

#Jyotisha #Trigonometry #Maths #HistoryOfMaths #MathHistory