This is a Q&A from the site Ask a Mathematician which I think is informative of the problem we’re facing in trying to understand the reality of physics in general, and perhaps General Relativity in particular.
Dear Mathematician, given Gödel’s theorem of incompleteness, is it possible for a complete theory of physics to come with a math that is complete, and still be true in all its statements?
I’m thinking the requirement of a complete formal system S to, by neccesity, include “gaps” could pose a problem for physics since they seem die hard on the math to be totally flawless.
For instance, the concept of singularity as an initial state pre-big bang seems rather accepted in most of physics, but in math it means “undifined” or “Dunno”. How akward it would be if the very fundation of every physical event, every cool equation and theory, could not be described by physics for as long as they (a) require the attachment of well defined math, and/or rejects the notion of a math saying “Dunno”.
Especially if that physical singularity was never broken, and therefore in effect still is a singularity. After all, logic has it that a true singularity has nothing external to it which can break or divide it, right?
To me, it seems reasonable that a theory in physics, supposed to cover Everything, must be unable to cover itself. That is if we a priori assume the theory to actually exist as an aspect of this Everything. Were it “outside” of Everything, it could get the complete picture, but that would question its ecological validity I guess.
Isn’t this the actual physics of Gödel’s brilliant idea concerning self-reference? So, complete math = incomplete physics and incomplete math possibly complete physics?
Then the answer
Most physicists have a healthy understanding of where math sits in relation to physics: if it works use it.
For a physicist, singularities don’t mean “the end of science” they mean “try something else”. There’s a post here that talks about singularities in physics.
Physics can be described very well using math, and every math system is incomplete, so ultimately we can expect that there are likely to be things about the universe that are likewise true-but-unprovable.
Hope that helps!
– If we don’t know what we’re working on, how can we tell if the math is actually working? Of course, in applied physics, as in engineering, this is a valid statement. That’s the pragmatic perspective.
But if we are about to hack the foundation to “what works”, stopping at “what works” is not good enough.
– If singularities means “try something else” to a physicist, that means s/he must disregard the Penrose Hawkins Theorem
proving singularities are essential to General Relativity. If the notion of singularity shows up, the physicist is encouraged to “try something else”. That seems an awkward approach, to disregard the very core of General Relativity.
– Wouldn’t it be a creative challange to figure out if the incompleteness of math, the incompleteness of physics and the fact that human cognition is based on relf-reference in some way are related to the nature of singularity? Wouldn’t it be nice to have a math that also was self-referent?
The last point about self-referent math is obviously a paradox. After all, the power of math is more like the opposite to self-reference. Math is designed for correspondence with objects that are not mathematical. If self-referent, math would probably end up entangled in circular functions that says nothing about the reality it is supposed to describe.
Perhaps that’s not a problem? Actually, that’s what I assume to be a possible way out of incompleteness in theory. Try this thought:
A singularity can be pictured mathematically as .5 + .5 = 1
As such, a singularity is both 1 and not 1.
It is not 0, nor is it 2.
It is 1 integer and 2 fractions.
It is both 1 absolute and 2 relatives.
It is of two faces where one face is Dual and the other face is Singular.
The Single face is same as the Dual face.
The Dual face is of Sameness united.
I suggest we do not try “something else”, but that we try harder to be creative with what we’ve got.
What we’ve got is 1. That’s the smallest quantity of unification, perhaps the only possible.
Math begins with 1 and not fractions. Without 1 in the first place, there is no one from which fractions can be measured and counted.
So while math works fine in our current universe, we can assume the post initial state singularity to be correctly described by the use of 1. If it wasn’t, then math would not correspond as well as it obviously does. Assuming that leads us to contemplate in what way This One
can be understood as equal to Those Halves
. We must be careful not to analyse the .5’s as if they were 1 divided in 2.
The equation here does not
say 1/2 = .5
It says that if we do it backwards, beginning with the current standard of 1, then we end up missing half the initial point of singularity. This is what we normally do, and that’s why we end up in uncertainty
I am saying that we must avoid breaking apart what has once been unified, or we will lose a vital aspect of reality as it is. Instead we should ask ourselves what halves would be required as to be the same
as 1. The trick here is to resist minds habit of manipulating the data as to build minds own understanding of it. Mind has a strong tendency to mean
everything and every thing. It cuts up input and conceptualize it as either this or
1 or 0
Big or small
Here or there
Dimensional or nondimensional
Absolute or relative
Objective or subjective
Particle or wave
Position or velocity
Discrete or continuus
Right or wrong
Cause or effect
Finite or infinite
Body or mind
Local or nonlocal
Space or time
Electric or magnetic
Singular or dual
Self or no-self
Bounded or free
Surface or bulk
X or Y
Real or imaginary
Me or you
… ad infinitum
This is the requirement for intelligence to reason about reality. It has to do this, or it cannot tell one from another one. No definitions are possible without this a priori reconfiguration of input, and without definitions we can not reason verbally/intellectually at all. Intellectual discourse is impossible without separating This from That. In order to enable gradients, mind also operates in opposites/polarities. By that, it can picture a scale with 2 extreme values and then place anything related to these extremes somewhere in-between. Can you imagine science or philosophy conducted without this being done?
If you can, please leave a comment and tell my how.
To round this brief pointer off, I will borrow from one of the truly great minds a few quotes that might perhaps at least some air of credibility to the above. Not as in trying to use Henri Poincaré as a proof of me being “right”, but to show the mindset that must be cultivated if we are to make progress in our shared understanding of what This is. The minds are just means to the end of knowledge.
Analyse data just so far as to obtain simplicity and no further.
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.
Mathematics is the art of giving the same name to different things.