One can see why historians like to describe mathematics in terms of second-order philosophical, cultural, socio-political effects (a clockwork universe, a rationalist garden, etc) but surely this is putting the cart before the horse: the biggest impact of mathematics is in the first-order effects: mathematics made solid, as it were, in architecture, engineering and technology in general - much more so than the surrounding aesthetics.
I guess I'm just not buying tendentious (and unprovable) connections between advances in geometry and reshaping of the social order. That said, I will check out the book.
Mathematics was nowhere in architecture, engineering or technology before the renaissance. Heuristics, written or taught were the only things used. There were no formulas, even geometry, that was invented BC had no place in architecture of 13th century.
Euro-centric historic chauvinism never ceases to amaze. It is always worth a chuckle to hear for example how "Rome conquered the known world" and your bon mot here how "Mathematics was nowhere in architecture, engineering or technology before the renaissance." It is truly astounding.
I believe you misunderstood me. Take any architecture-engineering book before 14th century, and I mean go back to the dawn of man, and you won't even find Pythagoras theorem.
It was all collection of heuristics that worked and the collections were made from hundreds of years of trial and error.
Big buildings in Rome were all built with heuristics that had no formulas.
Euclidean geometry in architecture is a modern thing.
It's not merely that. It is the willful disregard for the fact that Rome itself was very much aware of Parthian and Sassanid Persia, India, and China. An entire city in Iran was built by Roman prisoners of war!
I sometimes wonder if the persistence of this willful disregard, as a pedagogical factor, plays a role in current geopolitical matters which finds the West in disarray as to how to respond to the reemergence of the "unknown world".
Even before Rome, these regions were not unknown. The Greeks even fought them, settled there.
One of my personal long term topics[1] to research is how these two hitherto separate schools of thought have influenced each other. The western and the oriental thought may have a common genesis.
When you say "before the renaissance", you mean "between the fall of Rome and the Renaissance, in Europe", right?
Because, for example, the ancient Greeks used mathematics in the design of siege weapons - you have to solve a cube root to work out how big the torsion springs should be in a ballista:
This is a hypothesis, no written text in architecture or engineering contains mathematics. It was all heuristics. Euclides geometry was not used in architecture before renaissance.
Without having read the book, I don't think they are discounting the first. Only that what you call are second order effects are actually much very important in the long run.
This becomes much clearer when you do a cross-cultural study of these effects. There are many poor countries, where the average person is roughly as mathematically/scientifically literate as a person in the richest countries in the world, but the philosophical/cultural/socio-political parts of those countries are much divorced from scientific truth.
Logic is usually considered as a part of philosophy. There are people in philosophy departments studying formal logic. I think all formal sciences should be considered as part of philosophy because they provide tools and language for the deepest ideas.
Not in my experience. All the mathematics departments with which I've been associated have had logicians.
Perhaps philosophy departments also have logicians, and perhaps they take different approaches, but certainly "Logic" is considered a sub-discipline of mathematics.
When I studied maths at Oxford the foundations of logic course was taught by the philosophy department. And the university offers a maths and philosophy course. I think it's fair to say that logic is where maths meets philosophy.
It's interesting that they overlook a similar shift in both mathematics and politics in the 1930s. It was at that time that mathematicians showed that truth is undefinable and no finite formal system can encapsulate all of arithmetic.
I guess I'm just not buying tendentious (and unprovable) connections between advances in geometry and reshaping of the social order. That said, I will check out the book.