There's no relevant physics without mathematics, one has to work out the math before trying to answer the 'why' questions. There are no short cuts. (Note: I'm also a theoretical physicist :) )
The math follows from the basic assumptions made about nature. Assume translational, rotational and time invariance, add the finite speed of light and some less obvious assumptions and all of classical physics (for those non-phycisists: this includes special and general relativity) follows.
Assume charged fundamental particles and EM follows. Assume quantized fundamental units and quantum mechanics follows. Lots of relevant physics can be described and shown with hardly any mathematics at all.
The idea that nature is mathematical in essence is a notion that primarily physicists entertain. They mistake intelligence for knowledge and philosophers, that have actual knowledge of the philosophical problems involved, know better. The idea that nature is mathematical in essence, and that we have discovered this fundamental essence, is both hubris and usually utterly devoid of any philosophical sophistication.
I don't think anyone is actually saying nature is math. In that strip you can see that the line continues rightward, though I don't know if that was intentional.
Math is just symbols that happen to have congruence with something deeper in reality. They're representations that can effectively substitute. Math just inherently works to explain nature and physics.
And as far as philosophers, well I wouldn't say it's far-fetched to call someone like Stephen Hawking a philosopher. The danger is the philosophers who misunderstand, or who ignore the advancements in mathematics and scientific knowledge we've been making.
Math is just symbols that happen to have congruence with\
something deeper in reality.
In which you already sidestep an enormous debate about whether such a thing as 'a deeper reality' exists. Nietzsche's 'God is dead' doesn't apply to the Christian God or Gods in general: it applies to every absolutist philosophy. Wittgenstein's work on the limits of language may have consequences for even what thoughts can possibly make sense. Their whole lineage, including people like Heidegger, Davidson and Rorty, is completely ignored by such an implicit, seemingly innocent, assumption.
The danger is the philosophers who misunderstand, or who
ignore the advancements in mathematics and scientific
knowledge we've been making.
Philosophers tend to have a pretty good understanding of the meaning of physics. It is taught to them at length, because of the importance of the scientific advances you refer to. It's certainly not the other way around. The fact is that most intelligent people are mostly unaware of the last 150 years of philosophy. The time before that is all they hear about, because it is pretty easy to explain. After that, it gets hard. However, those folks, among each other, just keep repeating the debates of back then, without realizing the advancements that have already been made. It's a damn shame. Even when you have progressed as little as I have, just to understanding how complicated these dualities are, discussing the Cartesian mind/matter duality with them over and over again gets like teaching yet another kid to integrate. They may be top-notch phycisists, but they simply don't know philosophy.
You said that philosophy is only limited by the need to be internally consistent.
Well, for one, religious philosophy is often not internally consistent. Same for eastern philosophy. And most of the history of western philosophy, for that matter. So even your one criterion doesn't work when you force it to be universally applied to everything that's ever been given the title of philosophy.
And that's the problem with your statement. It takes the broadest possible subject in the history of man and makes a generalization about it that at best applies to some small sub-set and at worst applies to absolutely nothing.
What you are talking about is the validity of a philosophical argument. But just about every discipline in philosophy is also concerned with the soundness of the argument.
Even things like ontology or metaphysics are concerned with correlating their arguments with the way things actually are in the world.
Otherwise, all philosophical discourse would occur in symbols. But instead, the factual accuracy of the argument's constituent premises is also important.
So epistemology, ethics, political philosophy, aesthetics, philosophy of law, religion, mathematics, and language... 100% of these things are concerned with much more than being internally consistent.
Subset's of philosophy are often defined by their individual constraints. Huge areas are devoted to various assumptions such as "the bible is literally accurate" or "how people actually behave" (descriptive ethics), but the only thing they all have in common is a drive for reasonable levels of consistency. Taken to an extreme you end up with Math in all it's glory, but a complete lack of such consistency results in meaningless gibberish.
Consider the amount of effort spent to explain evil in a world created by a good, perfect, and all powerful god. Or the winding paths to define "truth" without creating a tautology etc etc. IMO, Philosophy sit's in the middle ground between religion which is happy to make inconsistent things up, math which requires self consistency, and science which must be grounded in reality.
huh? define a philosophical constraint and i'll undefine it for you... i believe the import is that it helps us come to terms with our humanity, which has no formal definition, by definition
mathematics certainly informs philosophy, the most pertinent example that comes to mind is godel on the works of wittgenstein... if you look at philosophy post godel/wittgenstein it has become continental philosophy (analytically formed 'novels') and analytic philosophy (which informs the impact of logic on natural language)...
each of these activities carries as much or as little weight as you would like to give it... but conservative thinkers might say the same of even pure sciences - are we really better off with all the devices/distractions of modern life?
You could invert the phrase and still make a lot of sense: 'Nature follows from the basic mathematical assumptions.
It's not just that. It's also that a lot of physics can readily be explained and understood without mathematics. It's just that with mathematics, it's easier to demonstrate the consequences. It's important to remember that Einstein got at relativity by understanding the right assumption to be made.
Just to be clear on this: I fully agree that in advanced physics, you sometimes you have to blindly 'do the math' and can determine the physical interpretation of the result only afterwards. However, that doesn't mean that there is no physics without mathematics, it doesn't mean that there isn't a lot of physics that can be understood without mathematics and it certainly doesn't mean nature is mathematical in essence. Hell, it isn't even physical in essence, unless you can reduce our minds to physics.
The notion that 'nature is mathematical in essence' is not logically seperated from assumptions about the object/subject and mind/matter dualities (and about whether they are dualities in the first place) that I doubt most of the phycisists espousing these notions are willing to support, because they are rather incompatible with other parts of their worldview.
I like what you say, but I am conflicted. As I understood it, math is the language that physics uses for us to understand/manipulate reality. Just as one understand "coffee" without the word coffee, one can understand projectile motion without understanding derivatives -- however in the absence of a language to describe it, the concept remains difficult to build upon.
I do not understand about it not being logically separated from assumptions about dualities (Platonic forms?) and their respective worldview. Could you explain more please.
Some guy by the name of Hartry Field claims to have successfully generated a logical axiomitization of Newtonian mechanics, without using math, only qualitative concepts like "betweenness."[1] Math can extend this model and is consistent with it, but it's not necessary in principle. Of course, this does not mean that you don't need math in practice, in the same way that continuations are the only control structure you need in principle but in practice you use loops and function calls and switches or whatever. What it does mean, from a philosophical view, is that math isn't necessary the quintessential 'language of the universe' that we thought it was.
As far as advanced mathematical physics goes, our problem is that we stop being able to easily understand what the math represents when you go deep enough. A first derivative is understandable as velocity, the second as acceleration, and multiplying that by mass to get force has some intuitive notion. If you look in the middle of a bunch of scribblings, it's possible to put the equation into words and it will make sense and have a physical interpretation. Special relativity and the beginnings of quantum mechanics came about because we had intuitive concepts, and were able to fit math around them. In modern physics, the math has come first, and the intuition later or not at all. Quantum phenomena still have competing interpretations (probability fields, many-worlds), which may not even be mutually exclusive. Once we figure out what in the hell is going on, the math may prove to be unnecessary, but right now the math is all we have to go on, and we sort of have to treat it like a literal truth in order to get anything done.
Thanks. This is kinda difficult to wrap my head around though. I found this passage interesting:
"By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions."
I will admit that I have a problem when fictionalism requires that we regard "2 + 2 = 4" as false (also from wikipedia). All in all an interesting idea.
> 'Nature follows from the basic mathematical assumptions.'
This statement is more treacherous than you give it credit for. If we understood all those mathematical assumptions, wouldn't that be the death of free will?
One might make the statement that mathematics is the act of discovering god's laws, but godel makes even that statement treacherous because he proves god's laws can never be consistent.
my understanding of the modern philisophical take on mathematics is that it's just the invention of useful tools for manipulating logical entities, which can seem rather pedestrian to the layman - but will never deter a true mathematician
