March 17th, 2019, 03:16 PM  #1 
Member Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3  How to approach philosophy in isolation
I want to selfeducate myself in philosophy and I need advice on how and where to start. I want to be a specialist in differential geometry and relativity and also harness the power of logic and reason to put sound philosophical arguments together about what's around me. I'm hoping someone could provide me with a "starter package" from their experiences to get me going.

March 17th, 2019, 03:22 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,838 Thanks: 653 Math Focus: Yet to find out. 
You might be struggling to begin your journey because the scope of your ambition is too broad. Refine it!

March 17th, 2019, 03:24 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,393 Thanks: 749  
March 17th, 2019, 04:04 PM  #4 
Member Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3  
March 17th, 2019, 04:06 PM  #5  
Member Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3  Quote:  
March 17th, 2019, 05:06 PM  #6  
Senior Member Joined: Oct 2009 Posts: 864 Thanks: 328  Quote:
It's a good idea to educate yourself in philosophy, and I highly recommend it. But if you also want some kind of expertise in relativity theory and differential geometry, you're going to need an entirely different toolset! Nothing that can't be done of course, but you seem to think relativity is a part of philosophy, while it really isn't.  
March 17th, 2019, 05:24 PM  #7  
Senior Member Joined: Oct 2009 Posts: 864 Thanks: 328  Quote: I just know what's going to happen. The OP will read this link. The OP will read several related links and papers. The OP will spend years studying this, and after 5 years he'll come back and still not know the mathematical definition of a manifold. The OP has then, in fact, learned nothing. Epistemology of geometry (and of math in general), is an interesting field. But I stress that it cannot be done without a good knowledge of geometry. This will entail that you read mathematics books written for mathematicians. Once you get the mathematics down, you should start doing philosophy of the mathematics. There are many philosophers who talk about mathematics without studying it indepth. Those always amount to nothing. Likewise, there are many philosophers who DID spend time looking into the rigorous math. Those are the ones who know what they're talking about. David Malament comes to mind as the prime example in the field of relativity. I could easily provide the OP with a roadmap to learning geometry, keeping in mind that he's interested in philosophical aspects. The very first book he should read would be Euclid's elements, for example. Geometry as a field really started there. Many questions in geometry, many advances in geometry and many unsolved problems in geometry originate eventually from the Elements. Reading it is absolutely necessary as a first step. Of course, it should be read with good commentary to back it up. A commentary that shows the flaws in the elements and how the arguments in the elements influence modern day thinking. Then the focus would shift to nonEuclidean geometries, such as hyperbolic and spherical geometries. Projective geometry would need to be covered. The Kleinian view of geometry would need to be covered indepth. Once the OP has the prerequisites, he would need to start differential geometry in R^n, basically from Do Carmo's book or similar. Once that is done, he will want to abstract away the ambient space and cover manifolds and topology. Somewhere down the line, he'd need to cover conics, group theory and its application to symmetry, geometries arising from a field (geometrical algebra), and perhaps a bit of algebraic geometry but probably not too much. Then he also shouldn't neglect physics, which had a very profound influence on geometry: Lagrangian mechanics, relativity theory, etc. Yes, I really deeply feel this is all very necessary to cover if you want to say anything meaningful about the philosophy of geometry. The geometers back in the day often happened to be great philosophers too, the prime example being Descartes. Studying their math is necessary in order to study their philosophy. The converse is true too. You should be somewhat well versed in philosophy in order to understand parts of math. For example, Gauss discovered hyperbolic geometry, but never dared to publish it for philosophical reasons. One should know about Kantian philosophy in order to really understand why hyperbolic geometry was so controversial back in the day!  
March 17th, 2019, 06:31 PM  #8  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,272 Thanks: 942 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Well said. Dan  
March 17th, 2019, 11:47 PM  #9  
Senior Member Joined: Oct 2009 Posts: 864 Thanks: 328  Quote:
But it also requires a lot of knowledge of philosophy actually! In a sense, SR is trivially wrong, since it is superseded by GR. Nevertheless, SR has a very wide range of applicability and is absolutely required to understand if you want to master GR! But this raises a very different point. What does it "mean" that special relativity is wrong, what does it mean that SR is right? These are philosophically very thorny issues. You could assert that SR is right because it agrees with experiments. But in that same sense, the epicycle explanation of the planetary orbits was also right, since it did predict the orbits to very good accuracy! It is simply a fact of life that any physical phenomenon can be explained by various different theories, some more elegant than others. Both epicycles as Newtonian theory predicts the planetary orbits, for example. Which one is "right", which one is "wrong"? It's not always clear cut...  
March 18th, 2019, 04:56 AM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
No physical theories are "right", except perhaps on a very superficial level. The best ones are merely the least wrong. Every mathematical model carries assumptions and simplifications that render them no better than approximations.


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