What's up with the Galileo hate? Even if he couldn't derive the area of a cycloid, doesn't give justification to condemn a whole scientific career (Galileo is the most overrated figure in the history of science?!). Shouldn't Galileo be measured what he did solve rather than what he didn't... failing one problem is hardly proof of general incompetence.
Besides, he's not really known as a mathematician but more for his works in physics, and he certainly isn't considered one of the great mathematicians of his time.
Just a few things we owe Galileo in physics:
* The principle of relativity. You might think that was Einstein, but the first theory of relativity was by Galileo in his 1632 "Dialogue Concerning the Two Chief World Systems" (before Newton was even born!). Galileo introduced this idea with a brilliant thought experiment: He asked the reader to imagine being in a windowless cabin on a smoothly sailing ship. He argued that no experiment you could perform inside the cabin (dropping a ball, watching flies, etc.) could tell you whether the ship was at rest or moving at a constant velocity. All the laws of mechanics would behave identically. This is the cornerstone of classical mechanics. In the context of special relativity, Einstein "merely" added 'the speed of light is c' to the list of laws of nature that hold in all inertial frames. But the general way of viewing laws of nature relative as being invariant to motion was Galileo's (the principle of inertia), and essentially the starting point for Newtonian mechanics. It doesn't seem like the work of someone only able to fiddle around with scales.
* The Law of Falling Bodies: The discovery that the distance an object falls is proportional to the square of the time. The first truly modern mathematical law of physics.
* Detailed telescopic observations: Moons of Jupiter, Phases of Venus, Mountains on the Moon & Sunspots, etc.
Isn't a mathematician arguing that Galileo was a bad scientist because he wasn't as good at deriving the area under some function as Archimedes, a bit like a fish arguing that Galileo was a bad scientist because he couldn't swim as fast as a tuna?
Archimedes really is an underappreciated figure. His ideas, specifically about calculating the area of shapes, already preempted the idea of the integral almost two millennials before Leibnitz and Newton.
When reading about ancient Greek mathematics it always is striking how little it resembles the mathematics taught in schools and how much it resembles the mathematics taught in University.
IIRC, the ancient Greek mathematics we learn about today was the university-equivalent mathematics of that era. Common people did not use geometric abstractions to figure out math problems. Before Fibonacci brought algebra to Europe, everyday calculations were done on an abacus. If no abacus was nearby, people emulated one by placing stones in lines on the ground.
Pre-university schools, even today, focus on teaching practical math. Most people can get by just fine without skills in abstract math, theorizing, and proofs (though those skill would make a lot of people much better at whatever they do).
My point was that we should consider how advanced the Greeks were with their understanding of mathematics, especially their desires for proofs. And we should contrast that with how mathematics is taught today.
It's obvious that "practical math" has always been the most important and first skill to teach. But that ends at basic trigonometry.
Students are learning how to do integration in highschool (not exactly a relevant skill), long before they are confronted with the idea of proof in mathematics.
It's also interesting to note that the thought experiment is actually plain wrong, unless you consider general relativity a given.
Galileo's argument is that the theory where heavier objects fall faster is inconsistent a priori, because affixing a small stone to a larger stone would cause the composed object to fall faster than the smaller stone was falling when it was free. However, there is no logical contradiction here: what could happen is that the combined object would have an acceleration that is the (weighted) average of the acceleration of the components - slower than the lighter object but faster then the heavier object.
In fact, this is exactly what happens in an electric field: if you have two objects with the same mass but different negative charge moving towards a large positive charge, they will accelerate at different rates (the one with the bigger negative charge will "fall" faster). If you then tie the two objects together, you'll get a combined object that has more mass and more charge; the total electric force will increase, but its larger total mass will mean that it accelerates less. Alternatively, you can explain it as the less charged object dragging the heavier object down, such that the combined object moves at an average of their speeds.
The fact that this doesn't happen with gravity is a very special property of gravity, that only experiments can prove. A priori, gravitational mass/charge could have been entirely unrelated to intertial mass, just like electrical charge. Only much later, with Einstein's general relativity, did we get an explanation of gravity that makes this more than a coincidence - and it turns out that gravity is not a force at all, at least not one that acts on objects.
- The blog offers no primary evidence for Descartes’s having a proof.
- Scholarly histories, based on critical assessment of surviving letters, treat the solution of the area problem as due to Roberval (and independently Torricelli) rather than to Descartes.
- The more carefully vetted sources place Descartes in the position of reacting to, or endorsing, Roberval’s result but not of originating it.
Therefore, the weight of evidence supports that the historical consensus is correct — Descartes did not solve the area under a cycloid; the blog's claim is likely an overstatement or misinterpretation."
What's up with the Galileo hate? Even if he couldn't derive the area of a cycloid, doesn't give justification to condemn a whole scientific career (Galileo is the most overrated figure in the history of science?!). Shouldn't Galileo be measured what he did solve rather than what he didn't... failing one problem is hardly proof of general incompetence. Besides, he's not really known as a mathematician but more for his works in physics, and he certainly isn't considered one of the great mathematicians of his time.
Just a few things we owe Galileo in physics:
* The principle of relativity. You might think that was Einstein, but the first theory of relativity was by Galileo in his 1632 "Dialogue Concerning the Two Chief World Systems" (before Newton was even born!). Galileo introduced this idea with a brilliant thought experiment: He asked the reader to imagine being in a windowless cabin on a smoothly sailing ship. He argued that no experiment you could perform inside the cabin (dropping a ball, watching flies, etc.) could tell you whether the ship was at rest or moving at a constant velocity. All the laws of mechanics would behave identically. This is the cornerstone of classical mechanics. In the context of special relativity, Einstein "merely" added 'the speed of light is c' to the list of laws of nature that hold in all inertial frames. But the general way of viewing laws of nature relative as being invariant to motion was Galileo's (the principle of inertia), and essentially the starting point for Newtonian mechanics. It doesn't seem like the work of someone only able to fiddle around with scales.
* The Law of Falling Bodies: The discovery that the distance an object falls is proportional to the square of the time. The first truly modern mathematical law of physics.
* Detailed telescopic observations: Moons of Jupiter, Phases of Venus, Mountains on the Moon & Sunspots, etc.
Isn't a mathematician arguing that Galileo was a bad scientist because he wasn't as good at deriving the area under some function as Archimedes, a bit like a fish arguing that Galileo was a bad scientist because he couldn't swim as fast as a tuna?
Archimedes really is an underappreciated figure. His ideas, specifically about calculating the area of shapes, already preempted the idea of the integral almost two millennials before Leibnitz and Newton.
When reading about ancient Greek mathematics it always is striking how little it resembles the mathematics taught in schools and how much it resembles the mathematics taught in University.
IIRC, the ancient Greek mathematics we learn about today was the university-equivalent mathematics of that era. Common people did not use geometric abstractions to figure out math problems. Before Fibonacci brought algebra to Europe, everyday calculations were done on an abacus. If no abacus was nearby, people emulated one by placing stones in lines on the ground.
Pre-university schools, even today, focus on teaching practical math. Most people can get by just fine without skills in abstract math, theorizing, and proofs (though those skill would make a lot of people much better at whatever they do).
My point was that we should consider how advanced the Greeks were with their understanding of mathematics, especially their desires for proofs. And we should contrast that with how mathematics is taught today.
It's obvious that "practical math" has always been the most important and first skill to teach. But that ends at basic trigonometry.
Students are learning how to do integration in highschool (not exactly a relevant skill), long before they are confronted with the idea of proof in mathematics.
I was going to say this is BS, and that Gaileo's big achievement was not undermined by this argument.
I then found that what I was going to argue was his big achievement was not as original as I had thought: https://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pis...
On the other hand he still seems to have made a significant contribution to laws of motion in his writing, but I am not sure.
It's also interesting to note that the thought experiment is actually plain wrong, unless you consider general relativity a given.
Galileo's argument is that the theory where heavier objects fall faster is inconsistent a priori, because affixing a small stone to a larger stone would cause the composed object to fall faster than the smaller stone was falling when it was free. However, there is no logical contradiction here: what could happen is that the combined object would have an acceleration that is the (weighted) average of the acceleration of the components - slower than the lighter object but faster then the heavier object.
In fact, this is exactly what happens in an electric field: if you have two objects with the same mass but different negative charge moving towards a large positive charge, they will accelerate at different rates (the one with the bigger negative charge will "fall" faster). If you then tie the two objects together, you'll get a combined object that has more mass and more charge; the total electric force will increase, but its larger total mass will mean that it accelerates less. Alternatively, you can explain it as the less charged object dragging the heavier object down, such that the combined object moves at an average of their speeds.
The fact that this doesn't happen with gravity is a very special property of gravity, that only experiments can prove. A priori, gravitational mass/charge could have been entirely unrelated to intertial mass, just like electrical charge. Only much later, with Einstein's general relativity, did we get an explanation of gravity that makes this more than a coincidence - and it turns out that gravity is not a force at all, at least not one that acts on objects.
I know this is polemical article is in good fun but chatGPT gives me the impression Descarte should not be counted with the others:
https://chatgpt.com/share/68dd758f-2bc0-8008-955d-a7dbd89399...
" Given:
- The blog offers no primary evidence for Descartes’s having a proof.
- Scholarly histories, based on critical assessment of surviving letters, treat the solution of the area problem as due to Roberval (and independently Torricelli) rather than to Descartes.
- The more carefully vetted sources place Descartes in the position of reacting to, or endorsing, Roberval’s result but not of originating it.
Therefore, the weight of evidence supports that the historical consensus is correct — Descartes did not solve the area under a cycloid; the blog's claim is likely an overstatement or misinterpretation."
I never understand peoples' desire to copy paste their slop into a comment.