WarpDrive’s Reviews > Mathematics of Classical and Quantum Physics > Status Update

I often envy you. I wonder sometimes why I chose to study philosophy, of all things, as opposed to this stuff that you're delving into. I often wonder just what my subject matter is, and I imagine I wouldn't have the same problem were I studying a scientific discipline. Sigh.

Elena wrote: "I often envy you. I wonder sometimes why I chose to study philosophy, of all things, as opposed to this stuff that you're delving into. I often wonder just what my subject matter is, and I imagine ..."

You shoudn't, Elena. I can honestly tell you that deep philosophical analysis can be as complex and intellectually challenging as any higher mathematics you may come across. They just work through a different modus operandi - a different conceptual framework.
Just as an example: I still learn new insights when I re-read Kant, and I feel that I have to go a long way before I can get to anywhere near your level of understanding of his philosophical import.

WarpDrive wrote: "Elena wrote: "I often envy you. I wonder sometimes why I chose to study philosophy, of all things, as opposed to this stuff that you're delving into. I often wonder just what my subject matter is, ..."

...and philosophy is something which I always found priceless in helping me begin to understand - or help me decide - what human existence is about.....

Grass is always greener, I guess!

Elena wrote: "Grass is always greener, I guess!"

perfectly said :-)

I do agree that this stuff can be extremely challenging, but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it tries to expand itself into when it moves away from empirical inquiry into purely conceptual analysis. So the contortonism and the agony and the ceaseless complication might, in this case, stem from the fact that we're operating in a vacuum. (Kant made that observation too). That is a depressing possibility, for me. Just think of how elaborate a fictional universe - say, that of Sherlock Holmes - can get. It can be extremely intellectually challenging just to figure out what came first and why in a fictional work. And don't even let me bring up alchemy or astrological system building....

Point is, once you're so divorced from any empirical inquiry, you're liable to get depressed and claustrophobic now and then. You get Hume's panic that maybe his philosophical reflections got him stuck in some kind of artificial abyss.

Elena wrote: "I do agree that this stuff can be extremely challenging, but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it t..."

I perfectly understand and feel the same, Elena, very true and honest points indeed.
I can only add that there are areas where you have to somewhat necessarily, and at least temporarily, divorce yourself from empirical inquiry: areas where science is so immature that, if you want to run any inquiry at all in order to respond to our human thirst for knowledge and meaning, some sort of conceptual analysis, as prone to error as it may be, appears to be the only realistic avenue.
Some people may say that, if some question is not amenable to empirical inquiry, then we should not delude ourselves into trying to answer it, and just state that we do not know. I tend to adopt a different stance - pragmatically, I think that we have at our disposition a tool-set comprising many different instruments - science, philosophical analysis, even art and poetry.
Different questions can require different tools - yes, in some cases, it feels like world-building or operating in a vacuum, but I think it is a price that I am personally ready to pay, as long as we are honest about the reliability of each tool (including science, which is far from perfect and far from capable of answering all categories of questions, at least for now).

...but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it tries to expand itself into when it moves away from empirical inquiry into purely conceptual analysis.

You will excuse me for adding what little I can bring to this discussion, but, with all due respect, I believe that you are incorrect in supposing that this is not mathematics of an empirical nature. In fact, a great deal of this, judging by the table of contents, is somewhat basic university mathematics, by which I mean undergraduate level. Forgive me for bringing up the Dark Ages, but I took my only course in functions and matrices (sic) as a second semester freshman. Multivariable calculus is a 3rd year course, or late second, and, admittedly, can be a bitch unless one is well grounded in calculus,(freshman year, 2 semesters, hopefully, for differential and integral.

But the point i wish to make here is that Physics students are applying this mathematics to solve hypothetical systems which are very much the same as any empirical analysis, albeit in a world which we do not directly see. However, I argue that this is no different from a world which we do see except that philosophy seems to attract every ding-dong who has managed to pass a course on Eliot's poetry, Intro to Phil, and has an extremely high opinion of his own self-importance. These people don't read any philosophy, except in novels, of course, but pontificate on technical things, invariably without knowing the details. Good philosophy seems to me no different than string theory in that both suppose a world where certain interactions occur according to formulae. they must both be formidably consistent, but neither may actually refer to anything from the world.

To the original statement. Physics is really coming back to earth after being out in Never-Never Land which is often mathematics. I am not suggesting that it is not theoretical in its constructs, but it doesn't appear to me that these constructs are any different from the construction of the language and schemata which are created in order to explain the vision which is philosophy.

The advantage concerning Physics is that without some background in mathematics, one would be hard pressed to speak of Hilbert space. (the one exception was when Michel Houellebecq makes a joke about marriage being like Hilbert space vector.) “What I’m trying to say, “Michel went on enthusiastically, “is that from an ontological point of view, the pair can be assigned a single vector in Hilbert space. Do you see what I mean?”

Heh, I appreciate the comments you both make, and I had them coming to me. I guess we learn the habit of self-flagellation when we study philosophy. We're so used to being pummelled by folks in other disciplines that we learn to pummel ourselves. Of course, there's truth in the notion of philosophy as a mother-discipline which generates a great deal of the theoretical suppositions of the sciences, arts, and of history. And I am sorry for randomly venting on this thread. Bad brain day.

Rhonda wrote: "...but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it tries to expand itself into when it moves away from emp..."

Hi Rhonda, just clarifying the contents of this book: it is actually, as per author's introduction, mostly at post-graduate level, not at undergraduate level.
We are talking here about, for example, advanced perturbation theory, radial Green's functions in 3 dimensions, Bessel's functions, Laguerre, Hermite and Legrenge polynomials, integral representations, dispersion relations, advanced theory of analytic functions, separable kernels, iterative technique in Banach spaces, ect.
I have done maths at undergraduate levels and most of such subjects were not addressed at all at the depth detailed in this book. By the way, yes multivariate calculus is treated in the book, but only as a quick recap - the author actually clearly assumes that the reader is already proficient in multivariate calculus (which is typically done at the beginning of year 2 at maths undergraduate level - well at least that was my experience, I think it also depends on the actual institution). Moreover, pretty decent knowledge in partial differential equations is also assumed by the author.
Complex mathematical problems (such as the theory of diffraction, one of the most complex mathematical topics in classical physics) are also treated.
Curiously enough, though, the author un-necessarily (in my opinion) spends some (too much) space on linear algebra, which I would have thought is normally already treated quite comprehensively at undergraduate level (still from my personal experience, though).
The author explicitly states in the intro that he assumes, as a pre-requisite, a standard undergraduate knowledge in both physics and mathematics. This text has been designed and used as a companion to graduate courses in mathematical physics - although the author also states that some parts (the easier ones, I guess) of it might possibly be also used at advanced undergraduate levels.

Referring to constructs of science as opposed to philosophy, while I do see a significant role to be played by philosophical inquiry, I think that we should not ignore that there is an important point of difference: empirical/experimental confirmation.
This is a difference that confers to science a unique epistemological advantage over other forms of inquiry (without discounting the importance of such other forms of inquiry). They are constructs, but with a different level of grounding.
Science "works", in general terms: every time we switch on a computer, we take a plane, we use the GPS, or we take an MRI, we all implicitly acknowledge this.
Simply, there is no other form of inquiry that has historically being so successful and proven so reliable, and this has to be acknowledged.
I however completely agree with Elena's statement that "there's truth in the notion of philosophy as a mother-discipline which generates a great deal of the theoretical suppositions of the sciences, arts, and of history." Very true indeed, and well stated. Science used to be called natural philosophy for a reason.

Elena wrote: "Heh, I appreciate the comments you both make, and I had them coming to me. I guess we learn the habit of self-flagellation when we study philosophy. We're so used to being pummelled by folks in oth..."

My intention here was not to be right, but to suggest the empirical nature of the application of mathematics to physics. Similarly, good philosophy is that which bears itself out as a means of testing one's life against it. It is the curious issue of generational transition that appears to cause us to disconnect from what other philosophers attempt to show us. It may be that all reasoned opinion requires a certain perspective, the importance of which becomes lost to following generations, barring those who take the time to immerse themselves in its structure.
As far as philosophy being a mother-discipline, I should have to understand what you mean by this. Without a doubt, philosophy generates theoretical suppositions about the world around us, but it is also an artificial construct to suppose that these things are easily divided except in education.
The significant beauty in philosophy, as I understand it, is that it takes a look at things through new eyes and sees new ways of first, thinking about them, and then acting accordingly. If activity is not the ultimate goal, then philosophy remains stillborn.

WarpDrive wrote: "Rhonda wrote: "...but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it tries to expand itself into when it move..."
I appreciate the clarification from someone who has actually seen the book. Indeed, when textbooks are typically very expensive, this seems like a relative bargain.
My point was that the basics of the mathematics was at an undergraduate level and I did not mean that doing graduate work did not go far deeper into the application. Indeed, my effort was that it IS application of the theoretical to a practical system, often resulting in utilities which we use every day, as you cogently point out.
Mathematics was often called the queen of the sciences, but very often, those who wished to use mathematics to solve problems have had little respect for the art, often rendering another less complimentary name. Engineering, for example, wishes to solve problems and often uses mathematics as kind of a meat grinder by which solutions are given. I cannot count the number of times I have had to correct sloppy math in this regard because one was too anxious to arrive at a solution.
In summary, by suggesting that these applications in physics create usable functions in the world in the forms of electric lights and an MRI, you are making my point: physics attempts to not only say things about the function of things around us, but to create concrete things which are usable.
While it is true that science depends upon the empirical testing to create its results, even when this empiricism is beyond what we can actually see, I am boundless in my admiration for the growth of study in these areas for the last hundred years or more that has caused this magnificent creation of devices designed to make our lives better. However, what I am saying to Elena is that philosophy deserves that same rigor in creating ideas which may become practical, not only through self-consistency, but through practical application

Rhonda wrote: "WarpDrive wrote: "Rhonda wrote: "...but I do wonder whether a lot of the challenge doesn't come from the sheer fact that the mind cannot get any traction in this void that it tries to expand itself..."

Thanks for the clarification of your thoughts, Rhonda.
PS: regarding the book, I could not recommend it highly enough. As you correctly pointed out, it is inexpensive and I must say that this is the possibly the best post-graduate book in applied mathematics that I have ever encountered. Better than far more expensive books of the same type and subject. By the way, it has an amazingly low number of typos - thing that is regretfully becoming rarer and rarer with the latest books - I hate it when I have to waste my time trying to make sense of some typos, in order to make it sure that it is not me who is making the mistake :-). And some of the examples in the book (like that on diffraction theory) are just first-class. There are a couple of areas where mathematical rigour could have been better - this is an area where I can be a real zealot, so it might just be me - but overall this book is a real treat. The only real disappointment was the lack of tensor calculus, but for some reason this subject often appears to be kept as a more specialist, separate item.