johncarlosbaez

johncarlosbaez, 4 hours ago (edited 3 hours ago) to random

Robert Recorde introduced the equal sign in 1557. He used parallel lines because "no two things can be more equal". And his equal sign was hilariously looooooong.

This is from @mjd's excellent blog article:

https://blog.plover.com/math/recorde.html

and I recommend following him here on Mastodon.

It's fun to fight your way through Recorde's text, with its old font and spellings. But if you give up, @mjd has transliterated it:

Howbeit, for easie alteration of equations. I will propounde a fewe exanples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes "is equalle to" I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.

The only real mystery here is "Gemowe", which means "identical" and comes from the same root as "Gemini": twins.

In the same book Robert Recorde introduced the mathematical term "zenzizenzizenzike", but I'm afraid for that you'll have to read @mjd's article!

highergeometer, 1 day ago to random

Two thematically related papers

Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, "Projective Space in Synthetic Algebraic Geometry"

Abstract: Working in an abstract, homotopy type theory based axiomatization of the higher Zariski-topos called synthetic algebraic geometry, we show that the Picard group of projective n-space is the integers, the automorphism group of projective n-space is PGL(n+1) and morphisms between projective standard spaces are given by homogenous polynomials in the usual way.

https://arxiv.org/abs/2405.13916

=====

Matías Menni, "Bi-directional models of `Radically Synthetic' Differential Geometry", Theory and Applications of Categories, Vol. 40, 2024, No. 15, pp 413-429.

Abstract: The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model.
http://www.tac.mta.ca/tac/volumes/40/15/40-15abs.html

mjd, 13 hours ago to random

This is the world's first use of the modern equals sign, from Robert Recorde's 1557 book The Whetstone of Witte.

(Screencap from Internet Archive's scan of the book: https://archive.org/details/TheWhetstoneOfWitte/page/n237/mode/2up)

(I also wrote a blog post a couple of years back explaining what it says if you are interested: https://blog.plover.com/math/recorde.html)

johncarlosbaez, 22 hours ago to random

An epidemiologist having a category-theoretic revelation. My colleague Nathaniel Osgood, discovering how the process of converting stock and flow diagrams into causal loop diagrams can be captured by a left pushforward functor between presheaf categories. These two kinds of diagrams are both important in the modeling tradition called 'system dynamics', which is used in epidemiology as well as economics and other disciplines.

System dynamics:

https://en.wikipedia.org/wiki/System_dynamics

"Left pushforward" as a method of data migration:

David Spivak, Functorial data migration, https://arxiv.org/abs/1009.1166

johncarlosbaez, 15 hours ago (edited 14 hours ago) to random

Joe Gore did graduate work on musicology and is an expert on medieval harmony theory. But when he played guitar on the Tom Waits song "Goin' Out West", he got handed a cassette tape of Tom Waits singing while pounding drums in a steel shed - and Waits had him play over that on an electric guitar tuned a quarter tone flat.

A great interview! He describes how Tom Waits gets great music from players by pushing them way out of their comfort zone. And near the end he gets into some serious harmony theory! I know just enough to know he's not faking it. I like the idea that in Lydian, the sharp fourth functions as a kind of second leading tone - a half-tone below the fifth, just like the usual leading tone is a half-tone down below tonic.

https://www.youtube.com/watch?v=0_wqAcj6EBU

julesh, 18 hours ago to random

I just had an epiphany. The purpose of an review rebuttal is not to convince the reviewers to change their mind - which is obviously not possible except in vanishingly rare cases - it's to convince the programme committee that the reviewers don't know what they're talking about

dmm, 20 hours ago to math

Here's something I just learned: the lucky numbers of Euler.

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.

Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.

Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).

The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.

h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.

References

[1] "Lucky numbers of Euler", https://en.wikipedia.org/wiki/Lucky_numbers_of_Euler

[2] "Heegner number", https://en.wikipedia.org/wiki/Heegner_number

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1)", https://oeis.org/A003173

[4] "Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1", https://oeis.org/A003173

#luckynumbersofeuler #heegnernumber #euler #math #maths #mathematics

dmm, 21 hours ago to math

The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].

Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].

A few of my notes on this and related topics are here: https://davidmeyer.github.io/qc/galois_theory.pdf. As always, questions/comments/corrections/* greatly appreciated.

References

[1] "Heegner Number", https://mathworld.wolfram.com/HeegnerNumber.html

[2] "Splitting Field", https://mathworld.wolfram.com/SplittingField.html

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).", https://oeis.org/A003173

[4] "Ramanujan Constant", https://mathworld.wolfram.com/RamanujanConstant.html

#galois #gauss #heegnernumber #ramanujan #math #maths #mathematics

sprout, 1 day ago to random

As a student, I once walked into a philosophy department's library and found a small gem discussing the origin and naturalness of different Boolean connectives. I remember several options for implication being discussed, a connective we nowadays learn about and whose properties are taken without question.

The paper below on 'why negation' reminds me of that experience some thirty years ago, though as research goes this paper is written with some quantitative arguments instead of only qualitative ones.

https://www.journals.uchicago.edu/doi/10.1086/715140

johncarlosbaez, 6 days ago to random

Is there a chance that the physicist Oliver Heaviside was really Wolverine?

image/jpeg

johncarlosbaez, 3 days ago (edited 2 days ago) to random

When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just send people out to the desert to die.

Oh wait, you don't have to imagine! It's happening already:

"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.

European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."

The full story is here: https://archive.is/1tEgl

(1/2)

johncarlosbaez, 2 days ago to random

I want to read this book: A Darwinian Survival Guide. Sounds like a realistic view of what we need to do now. You can read an interview with one author, the biologist Daniel Brooks. A quote:

...

Daniel Brooks: What can we begin doing now that will increase the chances that those elements of technologically-dependent humanity will survive a general collapse, if that happens as a result of our unwillingness to begin to do anything effective with respect to climate change and human existence?

Peter Watts: So to be clear, you’re not talking about forestalling the collapse —

Daniel Brooks: No.

Peter Watts: — you’re talking about passing through that bottleneck and coming out the other side with some semblance of what we value intact.

Daniel Brooks: Yeah, that’s right. It is conceivable that if all of humanity suddenly decided to change its behavior, right now, we would emerge after 2050 with most everything intact, and we would be “OK.” We don’t think that’s realistic. It is a possibility, but we don’t think that’s a realistic possibility. We think that, in fact, most of humanity is committed to business as usual, and that’s what we’re really talking about: What can we begin doing now to try to shorten the period of time after the collapse, before we “recover”? In other words — and this is in analogy with Asimov’s Foundation trilogy — if we do nothing, there’s going to be a collapse and it’ll take 30,000 years for the galaxy to recover. But if we start doing things now, then it maybe only takes 1,000 years to recover. So using that analogy, what can some human beings start to do now that would shorten the period of time necessary to recover?

https://thereader.mitpress.mit.edu/the-collapse-is-coming-will-humanity-adapt/

dpiponi, 2 days ago to random

Despite Wildberger being a bit off the usual conventional paths in mathematics, he's influenced me to the point where every time I write a line of code using an angle I ask myself if I could use an alternative "rational" representation.

https://research.unsw.edu.au/people/professor-norman-j-wildberger

johncarlosbaez, 4 days ago (edited 3 days ago) to random

A while back @pschwahn raised an interesting puzzle here:

You can define a well-behaved cross product of vectors only in 3 and 7 dimensions. The 7d cross product is weird because it's not preserved by all rotations of 7d space. But very smart people have told us there's a way to get the group of 𝟯𝗱 rotations to act on 7d space while preserving the 7d cross product. In fact, you can do it while also getting this group to act 'irreducibly', meaning the only subspaces of 7d space preserved by this action are {0} and the whole space!

The puzzle is: 𝗰𝗮𝗻 𝘆𝗼𝘂 𝘄𝗿𝗶𝘁𝗲 𝗱𝗼𝘄𝗻 𝘁𝗵𝗲 𝗳𝗼𝗿𝗺𝘂𝗹𝗮𝘀 𝗳𝗼𝗿 𝗵𝗼𝘄 𝘁𝗵𝗶𝘀 𝘄𝗼𝗿𝗸𝘀?

I got stuck on this so I asked some of my friends, and now Layra Idarani has outlined a nice way to do it:

https://golem.ph.utexas.edu/category/2024/05/3d_rotations_and_the_cross_pro.html#c063185

Interestingly he doesn't actually give the formulas; he just tells you how to get them. So I will need to do some work to check his answer! If you want to help out, that would be great.

Layra said "The devil of the details is in the eating". I thought the proof was in the pudding. Now I'm hungry for devil's food cake.

johncarlosbaez, 17 days ago (edited 17 days ago) to random

Tolstoy: "Happy families are all alike; every unhappy family is unhappy in its own way."

Mathematics: "Real tori are all alike; every complex torus is complex in its own way."

To be precise, a 'n-dimensional real torus' is a real manifold of the form V/Λ where V is an n-dimensional real vector space and Λ ⊆ V is a lattice of rank n in this vector space. They are all isomorphic.

An 'n-dimensional complex torus' is a complex manifold of the form V/Λ where V is an n-dimensional complex vector space and Λ ⊆ V is a lattice of rank 2n in this vector space. These are not all isomorphic, because there are different ways the lattice can get along with multiplication by i. For example we might have iΛ = Λ or we might not.

And so, it's possible to write a whole book - and indeed a fascinating one - on complex tori. For example a 1-dimensional complex torus is an elliptic curve, and there are whole books just about those.