The Full Ptolemy
We now start the second part of this series: an in-depth look at the Ptolemaic system.
Set Theory Jottings 3. The Paradoxes
Frege added an appendix to volume II of his 1903 magnum opus Grundgesetze der Arithmetik (Foundations of Arithmetic). It began:
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
Filed under History, Set Theory
First-Order Categorical Logic 10
JB: Last time we saw how to get some laws of logic from two facts:
• right adjoint functors between boolean algebras preserve products (‘and’),
and
• left adjoint functors between boolean algebras preserve coproducts (‘or’).
Filed under Categories, Conversations, Logic
First-Order Categorical Logic 9
Filed under Categories, Conversations, Logic
First-Order Categorical Logic 8
MW: We’re reviewing hyperdoctrines, which are specially nice functors B: FinSet → BoolAlg. When we have such a functor, any map f of finite sets gives a homomorphism of boolean algebras, B(f). But we’ve seen this is a morphism and a functor. (“It’s a floor wax and a dessert topping!”) What do you think about the term “adjoint morphism”? It might help keep the two levels straight.
Filed under Categories, Conversations, Logic
From Kepler to Ptolemy 5
Origins of the Ptolemaic System
We’ve worked backwards from Kepler to Ptolemy. What inspired Ptolemy and his predecessors (Apollonius and Hipparchus) to come up with this scheme in the first place?
First-Order Categorical Logic 7
MW: John, it’s been eons since we last discussed First-Order Categorical Logic: not since September 2019! (I read a lot of Russian novels during the break.) But New Year’s seems like a good time to resume the tale.
JB: Yes indeed! It’s been a long time, and it’s mostly my fault. Let’s see if we can get back up to speed.
Filed under Categories, Conversations, Logic
From Kepler to Ptolemy 4
Speed Laws
In Ptolemy’s system, the point on the deferent moves uniformly as viewed from a point called the equant point, or sometimes just equant. The equant, the center of the deferent, and Earth all lie in a straight line, with the center midway between Earth and the equant.
Set Theory Jottings 2. Cantor’s Paradise
Cantor’s Paradise
No one shall expel us from the Paradise that Cantor has created for us.
—Hilbert, “Über das Unendliche” [On the Infinite], in Mathematische Annalen 95 (1925)
I used to believe these myths about the history of set theory:
Filed under History, Set Theory
Diagonal Argument 