Is Mathematics a Paradigm of Truth in General Relativity?

[Ittiandro]

I have little exposure to mathematics beyond high school algebra and physics, but I have become increasingly interested in physics, particularly its most towering achievements of modern times : Quantum Physics, as well as A. Einstein’s Special and General Relativity, the latter of a particular interest to me as a gate to cosmology and its philosophical implications.

I did extensive reading about them, mindful, though, that my lack of mathematical training wouldn’t, regretfully, allow me to go beyond a basic conceptual understanding.

I found that S.R. is fairly easy to grasp both conceptually and also mathematically, because it is based essentially on Pythagoras’ theorem. Also, Space and Time are integral part of our experience of reality and befall our senses, even though we do not perceive their relative nature due to our inability to travel at the speed of light. .

The General Relativity Theory, to the contrary, has remained utterly impervious, at least to me, even to the most vague conceptual grasp, because curved space does not befall our senses, its “ reality” resting upon an intricate web of mathematical constructs, which however well they may theoretically describe its properties, in the end do not prove that space IS curved in the same way that the relativity of Space and Time has been experimentally proved true beyond the theoretical framework of its mathematical language.

Basically, my question is : in a case like this, what does a physicist really understand of the universe once he understands the mathematics behind this particularly hypothesis ?

Some may see this as a meaningless, purely philosophical question, which can be dispelled once we get to understands the mathematical language .

There are not many “ philosophical” problems, if at all, when doing algebra or pure mathematics or even financial mathematics, but when we reach the outer fringes of reality, like in cosmology, physics does inevitably entail( and coalesce with ) some measure of philosophical speculation and questions like the nature of mathematics, its obects and the understanding of the physical world that we glean by mathematical reasoning acquire full legitimacy.

Indeed, these types of questions are part not only of a well established branch philosophy, the philosophy of science, but also, more specifically, of the philosophy of mathematics, both engaging people with a good knowledge of science and mathematics, in some cases at the professional or academical level,..

I am not questioning the paramount importance of mathematics, as a powerful tool to understand reality in all fields of scientific inquiry, particularly in physics.

Indeed, mathematics clarifies our thinking and allows to better frame our questions and answers, by moving away from the ambiguity of words and verbal discourse ..

Without mathematics, Albert Einstein’s thought experiments on light never would have blossomed into the fully articulated theory of Special Relativity.

Ultimately, however, what is TRUE of the S.R. theory is not( or not only) its mathematical layout, but the reality to which mathematical reasoning has been applied, i.e. the nature of Space and Time themselves, a reality which is today fully validated experimentally ..

On the other hand, when we claim that Gravity is the curvature of space, I am no too sure if what we understand of the Universe in this particular instance through the use of mathematics is the Universe itself ( or some of its parts) in its causal links, or, rather, only the mathematical language itself.

If so, there would be a lingering suspicion of circular reasoning: the truth of scientific hypotheses stands or falls with their empirical validation and their predictive ability, but if we make mathematics the paradigm of truth ( in this case the truth that space is curved, based on the internal logical consistency of linear algebra and its basic elements, like matrices, vectors, etc) ) then we make a hypothesis( curved space) that is a priori not falsifiable: unlike causal explanations which are falsifiable by empirical evidence, mathematical constructs are universally true, (unless their internal links are broken by calculation errors or other contingent flaws), because number and the mathematical relationships among numbers are given A PRIORI as the very structures of our mind. We could not understand even such a simple arithmetic concept that 2+2=4 if we didn’t have the concept of number given A PRIORI in our mind.
Maybe somebody can comment on thisThanksIttiandro

 

[Dale]

which however well they may theoretically describe its properties, in the end do not prove that space IS curved in the same way that the relativity of Space and Time has been experimentally proved

This is a false premise. General relativity is very well proven experimentally. It is on a similar footing as special relativity.

we make a hypothesis( curved space) that is a priori not falsifiable

This is also false. GR is in fact eminently falsifiable.

Some may see this as a meaningless, purely philosophical question

That is correct. The mentors discussed and decided that this is too philosophical for PF. Based on that and the above false premises this thread will remain closed.