Reliability of physics as an exact description of the universe

[elemental09]

First, I'll briefly introduce myself, seeing as I just joined Physicsforums.com. I am a 2nd-year student at the University of Toronto in the physics specialist program, also pursuing a math major. I haven't actually been exposed to much physics in a formal setting since since my "real" physics courses only start this fall - up to now I've just done the introductory physics survey course and a bunch of math (currently taking ODEs and multivariable calc) - that is to say, forgive me if my knowledge is lacking in the fundaments (i.e. electrodynamics, classical and quantum mechanics formalism, stat mech, etc.).

On to the actual topic: lately I've gained an interest in the philosophy and underlying logical framework of math. Up until a few months ago, I always thought that the basics were entirely agreed upon by those well-versed in the field of mathematics - that is, the set theoretical bases, theory of proof, and principle axioms of mathematics were developed and studied enough to relegate new study of the bare basics to interesting quirks of little relevance or application (example: the formulation of fuzzy set theory). Then I discovered the constructivist school of thought. In this, some include the idea that for mathematical objects (like, for example, real numbers) must be constructable to be validly used when making inferences based on their existence. For example, the mean value theorem could be considered invalid simply because it postulates the existence of a real number without any reference to a method of constructing it. If one gives credence to this (albeit strict) interpretation of certain mathematical philosophies, then doubt is cast on the very existence of the real number system itself. If R is defined as the closure/completion of Q, using convergence of sequences, then there are (in terms of cardinality) "many more" irrationals than rationals, of which not all are constructable. Thus, in the strict interpretation, the very existence of the "continuum" (Rn) is invalid, and so calculus goes out the window.

I don't necessarily advocate this idea. But taking it into consideration for the sake of the argument, we can consider the implications for physics. Considering just about everything in physics is ultimately posed in terms of differential equations (from the most basic a(t)=v'(t)=x''(t) to general relativity), and that the very notion of differentiability (only definable on complete spaces like Rn) is now called into question, physics becomes at best a very good approximation. Any attempts to use physics, even in the most empirical sense, to explain ultimate mechanisms become questionable.

I could go on. I'm interested to hear others' opinions on the matter.

 

[Gear300]

Currently, we have no way of establishing that how we observe and analyze things could be taken as an absolute truth. We have no defined absolute truth.

 

[maverick_starstrider]

My number theory is intermediate at best but I don't see how the existence irrational numbers (or the fact that there are more irrational then rational) invalidates the dedekind cuts method of constructing the reals or the epsilon-delta proof of calculus. As for physics: Philosophically math and physics are disjoint. Math is an abstract construction and the assumption that a given mathematical model TRULY models a physical phenomenon is entirely unknowable in a metaphysical way. In fact all of science and predictions of reality are based on inductive reasoning which in math is not even considered valid since all math is deductive reasoning.

 

[elemental09]

Gear3000: you are right, can we not often tell when things are definitively not true, particularly in physics (e.g. the existence of aether)? What I am saying is that if calculus is unreliable, then many physical models that appear to hold water do so not because of valid logical inferences, and thus cannot be deemed to truthfully explain a mechanism.

Maverick: on the contrary, the existence of irrationals is fundamental to the construction of the reals. R can be defined (using Cauchy sequences - I'll have to read up on Dedekind cuts and get back to you; for now I assume the two methods are equivalent) as the set of all limits of convergent sequences of rational numbers. Clearly Q itself is a subset of this, and all the rest are irrational. My point in bringing these up is that, assuming constructability is a countable property, most irrational numbers are non-constructable. Regardless, the foundational theorems of calculus are certainly non-constructive (MVT, implicit function theorem, etc.) One can take this strictly to mean that any irrational number (actually any number at all!), without being explicitly constructed, cannot be assumed to exist for the purposes of proof. Lastly, differentiability cannot be defined on an incomplete space.
Regarding your comment on physics: it seems to me that the motivation for calculus' development was physics itself (certainly its invention was for this purpose). Physics has so intimately planted itself in the notion that space can be treated as having a bijective relationship with the reals that the implications of an ill-founded continuum scare me somewhat.

 

[maverick_starstrider]

yes, but once again, we can model physical behavior with mathematics and get the correct result, within error, every time we perform an experiment. That, however, would not be sufficient in a philosophical (or mathematical) sense to say that we have PROVED that our mathematical model IS the exact behavior of the physical phenomenon. Also, in regards to calculus, the original construction of calculus by Newton was nowhere near rigourous (in fact, the proof of the power rule assumes without any consideration that 0^0=1). This is why the limits proof of calculus was discarded in the 1800's (the bourbaki era) and calculus was reconstructed using the epsilon delta proof. In fact the notion of rigour predates the application of calculus to physics by about 200 years.

It is not the pervue of science to obtain a description of the universe in a philosophical notion of absolute truth sort of way. Indeed, anyone who wants to philosophically demonstrate that science is a pointless endeavour need only make reference to solipsism. To those people I say "you have fun buddy, I'll be over here inventing a longer lasting light bulb". The goals of science is to develop a pragmatic framework from which we can attempt to predict and control the world we perceive around us.

 

[maverick_starstrider]

By means of an example, often in physics we mathematically get results with multiple solutions and simply discard "unphysical ones" and our motivation for picking one model over another is correspondance with experiment. However, that being said, purely mathematical arguments have also led to profound physical discoveries (for example, the existence of anti-particles).

 

[mordechai9]

I don't at all see your point in regards to the "constructivist" objection. It seems to me that the irrational numbers have been shown to be constructable, theoretically, by the method you mentioned as well as the method of Dedekind cuts. Even if you can't construct the irrational numbers literally, why would you possibly negate the validity of the entire notion. That's comparable to saying that atoms don't exist because we cannot see them with our eyes.

Secondly, practically speaking, the real numbers are all constructable by hand, just give me a pencil and tell me which number you want. I will write it down for you and construct it right now. Certainly this isn't a rigorous mathematical construction in the sense of providing for the existence of some number which isn't rational, but does that remove its capability in representing exactly the same notion? We don't even compute (ever) with any truly irrational number, or unless I am mistaken, and someone out there is doing infinite sig-fig computations.

All of physics and even most of mathematics involved approximations, of a very healthy and expectable kind. Outside of logic we do not do proofs for 1,000,000 + 1,000,000 = 2,000,000, so someone could rightly claim that in most mathematics we approximate or assume these results based on sufficient prior information. But this doesn't make the statement circumspect at all, at least, not to anyone with sanity remaining. In physics and engineering, approximations ("ignore higher order terms!") are used so promiscuously with great success that it would be unfathomable to question their validity.

 

[maverick_starstrider]

Indeed, in physics the taylor series is your friend.