What does that mean? Physics is still empirical at the end of the day. Experiments decide what theories best explain the world. Math doesn't have such a requirement. It doesn't need to model natural phenomenon. Your physics lecturer sounds like a Platonist.
> "Your physics lecturer sounds like a Platonist."
I don't understand what this means, but it made me envision a McCarthy-esque witch hunt for "Platonist and Platonist sympathizers" lurking amongst the faculty
Platonist meaning an assumption that mathematical objects have a real existence, and the universe is inherently mathematical, so we can just defer to mathematical reasoning instead of observation. I'm applying the term in a modern setting, not Plato or Aristotle debating the forms.
This debate played out in String Theory were some proponents claimed physics had progressed beyond the need for observation in favor of beautiful mathematical reasoning, that provided great explanatory power. But String Theory so far has failed to deliver a theory which describes our universe. Physics still needs to explain the actual world.
As fundamentally opposed to Mathematical Plantonism, I am fully of the opinion that I am nowhere near anything but a minority position among those that hold an opinion on the philosophy of mathematics. It would be kind of hard to have a witch hunt for something so common amongst working mathematicians:
The saying that “the typical working mathematician is a platonist during the week and becomes a formalist on Sunday” is becoming increasingly familiar. During working days, they are convinced that they are dealing with an objective mathematical reality that is independent of them, and when on Sunday they meet a philosopher who begins to question this reality, they claim that mathematics is in fact the juggling of formal symbols (see Davis et al., 2012, p. 359). The Platonist attitude of the working (rather than philosophizing) mathematician is so common that Monk (1976, p. 3) was tempted to make a subjective estimate to the effect that sixty-five percent of mathematicians are platonists, thirty percent formalists, and five percent intuitionists. [1]
[1] - A Metaphysical Foundation for Mathematical Philosophy (Wójtowicz,Skowron 2022)
Just say witch hunt. McCarthy was entirely correct that there were a lot of communists and communist sympathizers, so many that many of the people he thought were helping him were themselves communists or communist sympathizers. Witches on the other hand are not real.
He missed most of the actual Soviet agents, though, right? Both seem to have mostly focused building up a lie to suppress some outside-the-norm element that the guy in charge wanted to hunt. That McCarthy also managed to have some real agents to miss is not a big difference.
So you claim it was a total coincidence he went after Soviet agents? The witch analogy here is that there are real witches but they are so powerful/well hidden witch hunters cannot find them and are diverted to patsies. This wouldn't mean witch hunting is bad, just that we needed better witch hunters.
I don't think OP is really wrong here. Wasn't there a debate in the late 19th century that basically asked if math had to have some mapping to the natural world, or should it work independently? I though there was some argument about this with Hilbert and Poincaré about this, and Hilbert more or less won.
Geometry? Lobachevsky actually proposed a test on measuring sum of angles of a celestial triangle to decide which geometry actually applied to the real world.
There are multiple geometries though, they don't have to describe the real world. In mathematics you are free to base your geometry on whatever axioms you want, whether they are realistic or not. I'd say the question of which geometry applies in the real world is more a question for physicists (or cosmologists, today).
No, it's trying to prove that Euclid's parallel postulate can be derived from other axioms is what goes back millennia. People were certain it's a) true, b) necessary consequence of other axioms. Gauss was probably the first to consider the possibility that it may be false; others at best tried reductio ad absurdum, arrived to some wildly unusual theorems, decided those were absurd enough to demonstrate the truth of the fifth postulate, and went back to trying to derive it.
It's either true as an axiom or true as derived from other axioms and/or theorems. In neither case does latching onto Euclid's other common notions/postulates/theorems as the selection it must be proved true from make sense as a 1.5kya long task.
I think there must have been a sense that it was true only as an axiom. Proving it from other axioms/theorems was then a goal to secure it's truth "further". But you'd only attempt that if you thought there was something questionable in the first place.
> But you'd only attempt that if you thought there was something questionable in the first place.
No, that's not the only reason. Come on, people actually wrote why they tried to prove it in their commentaries on Euclid's Elements.
For example, some people found that this postulate, compared with the first four, is not really that self-evident and also has a sudden jump in the complexity of its formulation. That's why some courses on geometry replaced it with something different (but equivalent), like "the sum of angles of any triangle is 180 degrees", or "for a line and a point not on it, there is exactly one line parallel to it that passes through that point", or "there are triangles with arbitrary large areas", etc.
The real world determines which geometry applies. That's the crucial difference. Sometimes physicists use mathematical reasoning to figure things out that map onto the world and later observation validates their reasoning. But observation can also invalidate, and then physicists have to go back to the drawing board or devise new experiments.
It's a very weak loose statement, but I think the idea is that leading scientific and mathematical thinkers (think Newton as a quintessential example) were "natural philosophers" who studied whatever caught their interest and took it wherever it went. Astronomers invented lenses and ground them and studied the starts and developed algebra and calculus to model the observations.
Some people were more narrowly focused, like Gauss who did mostly math (but an amazing breadth of math!)
There was a lot of hesitancy about math that couldn't be empirically illustrated by building out of atoms, like irrational numbers and then transcendental numbers and imaginary numbers and then infinite structures.
That's metaphysics not science. I'm not against metaphysics, but philosophy is distinct from physics. And it's Tegmark's particular metaphysics. Interesting but hardly a consensus among physicists or philosophers.
