Is Mathematics a Paradigm of Truth in General Relativity?

In summary: 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.
  • #1
Ittiandro
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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
 
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Ittiandro said:
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.

Ittiandro said:
we make a hypothesis( curved space) that is a priori not falsifiable
This is also false. GR is in fact eminently falsifiable.

Ittiandro said:
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.
 
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Related to Is Mathematics a Paradigm of Truth in General Relativity?

1. What is the role of mathematics in general relativity?

Mathematics plays a crucial role in general relativity as it provides the language and tools to describe and understand the complex concepts and equations involved in the theory. It allows for precise and accurate calculations and predictions of the behavior of matter and energy in the presence of gravity.

2. Is mathematics the only way to understand general relativity?

No, mathematics is not the only way to understand general relativity. While it is the most commonly used method, there are other ways to conceptualize and explain the theory, such as through visual representations and analogies. However, mathematics is essential for making precise predictions and testing the validity of the theory.

3. How does mathematics provide evidence for the truth of general relativity?

Mathematics provides evidence for the truth of general relativity through its ability to accurately predict and describe the behavior of matter and energy in the presence of gravity. The equations of general relativity have been rigorously tested and have consistently been found to accurately describe the observations and experiments conducted in the field of astrophysics.

4. Are there any limitations to using mathematics as a paradigm of truth in general relativity?

While mathematics is a powerful tool for understanding general relativity, it is not without its limitations. For example, the equations of general relativity break down at the singularity of a black hole, and our current understanding of mathematics is unable to fully explain this phenomenon. Additionally, there may be aspects of the theory that are currently beyond our mathematical capabilities to fully comprehend.

5. How does the use of mathematics in general relativity compare to other scientific theories?

The use of mathematics in general relativity is similar to other scientific theories in that it provides a framework for understanding and making predictions about the physical world. However, general relativity is unique in its complexity and the precision required in its mathematical descriptions. Many other scientific theories, such as Newtonian mechanics, are simpler and do not require the same level of mathematical sophistication as general relativity.

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