René Descartes invents calculus, joins empiricists, lives longer and more

[Hnau]

René Descartes
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I've been on a Descartes kick for the last few days and have been reading several of his works. I found it sad that his theories, which came to be known as Rationalism, were considered an opposing branch of science to the experimental methods being developed by Francis Bacon and others in England, which came to be known as Empiricism. There was a scientific feud in the 17th century over what methods were better, until finally Isaac Newton united the two with his Philosophiæ Naturalis Principia Mathematica which used mathematics (the domain of rationalists) to prove empirical claims (gravity and the orbit of planets).

What if we get the same Descartes, but the philosophy he develops is different? I have a very rough outline of what this entails, but it starts with him abandoning Le Monde early on in 1630 and moving immediately to his search for the "first causes". As such, we get Discours de la méthode in 1634, three years early, devoid of Les Météores and with a much shorter Dioptrique. Also, instead of moving from Je pense, donc je suis straight to the ontological argument for God, Descartes devotes an entire part of the treatise to proving the validity of the "blank slate" concept (tabula rasa) and how the mind creates ideas based on what it perceives. He then uses this to give further support for the ontological argument ("if the idea of God is never perceived, it must have been put there by God...") Even more divergent, before he goes into La Géométrie, Descartes explains why mathematics is present as a concept in the human mind by making what is essentially the evidentialist argument for where the idea of numbers and math come from (as seen in this video beginning at 8:11).

This is where Cartesian philosophy begins to diverge even more and where I could use some help on the specifics. I'd like to see Descartes come up with some equivalent, however more primitive, of John Locke's Essay Concerning Human Understanding which would expand upon Descartes' mention of Tabula Rasa and which would also put him firmly in the camp of the Empiricists. We avoid his stronger opinions on the Mind-Body Problem and so dualism never becomes a significant part of his philosophy. There are other things Descartes could produce ITTL that we could discuss, but what would be fascinating is if beyond what we have so far, he starts working on the development of calculus and becomes interested in planetary motion, which spurs him to write and publish an equivalent to Isaac Newton's Principia. I'm looking at the year 1648 as a possible date for such a work to be published. By formulating the laws of motion and proving universal gravitation, Descartes will be able to unite rationalism and empiricism before the feud really ever begins.

Let's take the story that René Descartes was poisoned by a Catholic missionary seriously. The theory actually seems very plausible. When he gets to Sweden to tutor Queen Christina in 1649, he's even more controversial than IOTL and this makes him more cautious. Descartes avoids assassination, has a much bigger impact on the queen, and dies of real pneumonia in the late 1650s.

The timeline that follows could be very interesting indeed. What would the other philosophers and mathematicians of the age do with Descartes providing an even greater foundation for further work? How would society be affected with a more unified scientific movement and an earlier discovery of classical mechanics? I also think it would be fascinating to see the already freethinking, rebellious Queen Christina become a deist of some sort instead of converting to Catholicism. What might happen if she keeps the Swedish throne? What if instead of involving Sweden in the Second Northern War, Christina decides to make important reforms to education and introduce religious skepticism?

What do you think about this admittedly very rough scenario?​

 

[Dathi THorfinnsson]

I wonder if Descartes was too good a mathematician to invent calculus. The initial formulations of Newton and Leibnitz had logical holes you could drive a truck though. Sorry, coach and four.

It wasnt until cauchy and the bernoullis and euler and company did a lot more work that calculus became acceptable math. Just sayin'.

 

[Hnau]

Interesting. As I understand it, though, it was Newton's at least rudimentary understanding of calculus that made it possible for him to figure out universal gravitation and the laws of motion, correct?

 

[Mongo]

One mathematician who was almost the same age as Descartes (two years younger) who did significant work on the foundations of calculus was Bonaventura Francesco Cavalieri.

Building on the classic method of exhaustion, Cavalieri developed a geometrical approach to calculus and published a treatise on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635). In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. Such elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri's method. As an application, he computed the areas under the curves – an early integral – which is known as Cavalieri's quadrature formula.

 

[Dathi THorfinnsson]

Interesting. As I understand it, though, it was Newton's at least rudimentary understanding of calculus that made it possible for him to figure out universal gravitation and the laws of motion, correct?

Oh, Newton had a very useful calculus, basically everything an engineer needs to know short of partial differential equations.

The problem was that his stated theory (using infinitesimals) was only demonstrated to be made rigorous in the 60s, or maybe z70s of the 20th century.

Basically, he did a lot of hand waving, and the results worked in normal cases.

Its entirely possible that archimedes, had he been willing to hand wave the theory could have had calculus back then.

Mind you, Newton, Leibnitz and Archimedes were all brilliant, brilliant men. Sometimes you need to say 'let formalism be hanged, this works'. Especially in physics.

 

[Hnau]

One mathematician who was almost the same age as Descartes (two years younger) who did significant work on the foundations of calculus was Bonaventura Francesco Cavalieri.

So maybe ITTL, instead of Newton and Liebniz fighting over who invented calculus, it'll be Descartes and Cavalieri? Does it sound plausible to recreate that feud here?

What's interesting is this mathematical flurry of activity will be taking place in Catholic strongholds, rather than Protestant ones. Though I'm not ruling out Newton and Liebniz from leaping even further using Cartesian calculus.

Oh, Newton had a very useful calculus, basically everything an engineer needs to know short of partial differential equations.

The problem was that his stated theory (using infinitesimals) was only demonstrated to be made rigorous in the 60s, or maybe z70s of the 20th century.

Basically, he did a lot of hand waving, and the results worked in normal cases.

Its entirely possible that archimedes, had he been willing to hand wave the theory could have had calculus back then.

Mind you, Newton, Leibnitz and Archimedes were all brilliant, brilliant men. Sometimes you need to say 'let formalism be hanged, this works'. Especially in physics.

Very interesting. So it seems plausible that Descartes could do this if he had simply been willing to do a bit of hand-waving. Nice.

 

[Hnau]

Queen Kristina of Sweden​

My mind gravitates to political divergences. Having read more of Queen Kristina, I've decided that she will still resign in the 1650s, but a year or two later. Instead of ending up in Rome, she lives for a time in the Low Countries, Germany, and Switzerland. Kristina never becomes a Catholic and without resources from the Church lives a little less extravagantly. She still tries to act as a patron of the sciences and the arts, but little of note happens.

This will of course leave massive butterflies to be dealt with by her successor Karl Gustav when he barrels headfirst into the Second Northern War. He still dies though around 1660-1662 from pneumonia on the battlefield somewhere.

This opens up a chance for Queen Kristina to take back her throne, as she never gave it up for Karl's son. Some thought she would do this in OTL. ITTL she does and takes to governing much differently than before. She introduces financial reform something like the OTL Great Reduction a decade earlier, and sponsors legislation that establishes the freedom of the press, and one of the most progressive laws yet on freedom of religion. A tax is put into place to pay for universal education mirroring the Scottish system. I can't say what she would do in foreign affairs, but I'm guessing she would keep Swedish forces out of superfluous wars. Kristina rules until she dies in 1690, leaving the throne to a different Karl XI.

Because of these reforms, there is a Swedish Enlightenment along with the Scottish Enlightenment in the 1700s. We see a whole bunch of philosophers, mathematicians, religious reformers and inventors come out of Sweden that didn't IOTL, mirroring the intellectual development of Scotland at the time.​