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Words to live by.
Turtles.
On Philosophy of Math: I see it’s been debated whether there is any philosophy involved in math, inasmuch as Math consists of solid reasoning based on uncontrovertible hard-core logic and axioms. Where the Philosophy comes in is, how important is all that logic and formal proving of things anyway?
I read an essay once, in which it was pointed out that prevailing attitutes about this have changed over the centuries. In Euclid’s time, Geometry was seen largely (just as today) as a mental exercise in formal logic and construction of proofs from axioms, definitions, premises, etc. That’s what you find Geometry textbooks full of, then and now.
But in Newton’s time, Math was largely the province of practical empirical scientists (often astronomers) who simply needed formulas to describe their observations. Proving things wasn’t such a big deal then. Note that Calculus, as developed by Newton and Leibniz and others over several hundred years, lacked the “epsilon-delta” concept of limits, and was thus unable to prove much of anything that went on in Calculus. It was only fairly recently (19th century?) that somebody came up with the ε-δ idea, whereupon it became possible to prove all the rules that had been in use since Newton and Leibniz. Before that, everyone seemed satisfied simply to note that the rules seemed to work, and seemed to give the right answers.
Today, mathematical rigor is once again very much in fashion. Why? That’s Philosophy of Mathematics.
See my experience here for discussion of what unconfined electrons do.
Quarks, as noted, come in many colors. I had a jar of those once too. We had a thread last October, in which we discussed, at length, techniques for counting them. (tl;dr: Learn some higher math.)
I never worried much about counting my quarks, at least until I started running low on them. But what do they DO? I found them good for curing the common cold and they were helpful for relieving flat feet too. Mixing some into my gas tank did absolute marvels to improve my mileage. Those were the more energetic quarks. But, like scruples, they are perishable, and you need to keep them refrigerated.
[QUOTE=Senegoid]
Note that Calculus, as developed by Newton and Leibniz and others over several hundred years, lacked the “epsilon-delta” concept of limits, and was thus unable to prove much of anything that went on in Calculus.
[/QUOTE]
The \delta-\epsilon formula of calculus is due to Weierstrass, from about the mid-18th century. I don’t think that supports your assertion, though. Newton, at least, certainly did provides proofs; they were just cumbersome geometric proofs and proofs by exhaustion rather than modern analytic proofs. Second, the subject recognizable as modern math only dates from about mid-18th century; before that, it was mostly subsumed into engineering and physics (where the more interest was more in calculation and applied results rather than rigorous proof), or else throwbacks to ancient plane geometry problems. Modern mathematics— groups, rings, manifolds, CW-spaces, category theory, schemes, etc.— is a fairly recent thing (and has very little to do with high-school or beginning undergrad math, up to about the level of calculus); before that, the subject was mostly about solving particular equations or differential equations. Comparing that to 17th century mathematics is like comparing chemistry to alchemy.
[QUOTE=Senegoid]
Today, mathematical rigor is once again very much in fashion. Why? That’s Philosophy of Mathematics.
[/QUOTE]
In fashion? You make it sound like an arbitrary whim, rather than an inherent part of the subject. There hasn’t been a reputable university for at least a century that would be content with vague, handwaving arguments instead of proof. If you want to talk about what constitutes a rigorous proof, that’s more a decision about mathematical particulars than any philosophy; at the very least, if I wanted to decide whether a proof was sufficiently rigorous and complete, I would ask another mathematican, certainly not a philosopher. More to the point, what would a philosopher of mathematics have to offer me, a mathematician? And if the point isn’t to make themselves useful in mathematics or to mathematicians, what is the point?
Understanding what it is that you do when you do mathematics. The philosophy of language isn’t geared to making people better speakers, and the philosophy of law isn’t (mostly) geared towards making better laws. Mathematicians may be interested in proving some long-standing conjecture, but philosophers interests are simply aligned differently; to expect to enlist their aid in mathematical endeavors is just to misunderstand philosophy.
I think it’s not that philosophy has no place in physics (and probably maths too), it’s that nowadays nearly all the philosophizing that helps push the understanding of the subject forward or offers original points of view on the subject come from physicist. The simple reason that the kind of broad, but in depth understanding of the subject matter needed to form original and innovative points of view are much more likely to be possessed by physicists.
Take the issue of particles: They can be seen as fundamental, trivial objects or as almost emergent non-trivial objects, depending on how you look at it as viewed through the lens of quantum field theory. This is because the actual connection between the quantum field and the concept of ‘a particle’ is, in a general sense, non-trivial. I.e. there is no known way to take an arbitrary quantum field and say this field describes n or <n> particles (where <n> is the expectation value as a quantum field doesn’t necessarily describe a fixed number of particles). Even when you can view the number of particles as a quantum mechanical observable and therefore have a set procedure to calculate <n>, certain kind of transformations which represent transformations between different reference frames can change <n> (i.e. two different observers can count different numbers of particles). If you view quantum field theory as fundamental then the concepts of particles are not very trivial at all for these reasons, but if you view it merely as a tool for describing quantum mechanical situations then the concept of particles are trivial (because they are a fundamental concept in QM).
Pressed “post reply” too early:
In order to really understand the issues around particles in QFT needs a very deep understanding (please note a kind of understanding that I do not even remotely possess myself when reading the above paragraph) of very difficult to understand subject. It’s these sort of issues that make it very difficult for philosophers, lacking the training of physicists to contribute to these sort of philosophical issues.
Turt…
… you win
I knew I should’ve posted that yesterday when I thought of it.
Turtles.
You can probably find some examples around the house. Like in the back of the freezer or in an old drawer.
Spam reported.
Well, but then, what makes a philosopher, or a physicist? Is it just what subject you happened to do your PhD in? I’ve read books about the philosophy of QFT by both notional ‘philosophers’ (Richard Healey, Paul Teller) and ‘physicists’ (Sunny Auyang), and while Auyang’s book certainly was the most demanding technically (and I’m not gonna pretend to have even understood half of it), I’m not sure that it was in any fundamental sense ‘better’, or even more competent, than the others.
Regarding QM, I think actually many advances, conceptual as well as technical, have philosophy (done by both philosophers and physicists) as their origin—without the philosophical engagement with it, probably few people today would talk about the Kochen-Specker theorem, and though Bell was a physicist, I would consider his famous theorem to be quintessentially philosophical in nature (in fact, I remember one quip about a senior scientist expressing his gratitude at having ‘lived long enough to have seen a metaphysical issue decided in the laboratory’ upon hearing of the first conclusive Bell tests). And obviously, this is really what motivated the study of entanglement, ultimately leading to modern quantum information science (I’m simplifying somewhat here, in case it wasn’t obvious). So without philosophy, I would be without a field! (Maybe that explains my pro-philosophy bias, though others in the field don’t really seem to share it.)
Aren’t the “interpretations” of quantum mechanics really a matter of philosophy rather than science? Science says that the equations of quantum mechanics can produce numbers which match the numbers we measure in experiments (thus the old “shut up and calculate” interpretation).
As far as I know, there is no possible experiment that can distinguish between the Copenhagen interpretation, the Bohm interpretation, or the many-worlds interpretation.
Someone once defined quarks as the dreams that stuff is made of.
The answers that the equations give must always match one another, but each approach is mathematically distinct and carries its own logic about what is happening to give those answers.
In a very simple analogy, if 120 is the answer, then the math to get to it can be thought of as 10*12 or 20+25+30+45 or 6! or 24,000/200 or 3[sup]3[/sup] + 99 - 60. They are all equivalent interpretations but you would visualize a different procedure along the route to the answer. Algebra gives you the same answers as geometry because they are also equivalent (or duel). You can never say that one is righter than the other. Lots of math works that way. You can proclaim that the processes are differing philosophies, but why would you do so?
The Dreams That Stuff Is Made Of: The Most Astounding Papers of Quantum Physics–and How They Shook the Scientific World is a 2011 book by Stephen Hawking. But it’s also a 2000 book, The DREAMS OUR STUFF IS MADE OF: How Science Fiction Conquered the World, by Thomas Disch.
I would use Bell though as a prime example of my point - yes, you could view his most important work as being philosophical in nature, but if he hadn’t had the, academically speaking, physics background it’s difficult to imagine that he would’ve been able to have those insights.
There’s a very good article in today’s Fermilab newsletter that touches on this in the course of discussing the equivalence of matter and energy.
Well, I wouldn’t describe anything at that level as “zooming around”, since our classical conceptions of “motion” don’t really apply at a quantum scale. I suppose it’s good enough for a layman’s treatment, though.
Incidentally, the Higgs mechanism, though apparently involved in the mass of the individual quarks, is not involved in the binding energy of the proton or neutron. So when people say that the Higgs is responsible for giving us mass, they’re mostly wrong.