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I'm reading Learning to Reason by Nancy Rodgers, and she says:

"Truth cannot be absolute in mathematics. Truth is relative, relative to the system that we construct for it."

"Today we consider an axiom to be nothing more than a sentence that is assumed true for a particular system. The same sentence could possibly be false in another system"

then she gives an example (you don't need to read it all, it's just an example):

The axioms for Euclidean geometry were based on the human visual perception of straight lines, but our visual perception is limited to very small distances. Our intuitive notion of straightness is based completely on light rays. Einstein predicted that a ray of light would be curved over large distances. Light rays are distorted by a gravitational field. The gravitational field comes from all the mass hanging out in that vicinity. The mass is hanging out there because of the curvature of space. An axiomatic system that models space as curved is different from Euclidean geometry. Even though we have curves in Euclidean geometry, space itself is not curved. Euclidean geometry is a good model of physical space when we are only concerned with small distances, when cosmic distances are involved, non-Euclidean geometry may provide a better model.

Now, I don't know if I'm misinterpreting things, but if "truth cannot be absolute in mathematics" does it mean that it can't be true "for everyone in the universe"? I've always thought about math like the realest thing someone can know, something that everyone in the universe must know (for example, if an ant doesn't have any food, it understands that it has $0$ food, so it knows about quantities).

So my question is, do these paragraphs mean that an axiom or a rule in general is not always true in every system, or it means that math is true only "in our heads" because we think of it in this way?

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cekami7844
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    You can bet that aliens have also discovered that nature of mathematics and the need for an axiomatization. You can also bet that they have discovered the same axioms as ours. Because the axioms we use are so "natural". –  Mar 16 '20 at 15:03
  • Different settings have different conditions they satisfy: some groups are commutative and some are not, some metric spaces are complete and some are not, some topological spaces are Hausdorff and some are not, and so on. When you know your group is commutative, then theorems about commutative groups apply to it, but if your group is noncommutative then theorems about commutative groups might not apply to it. There is nothing profound about that. Maybe if you model the real world with math you'll find a model only is a good fit within certain constraints, but that's not an issue of math. – KCd Mar 16 '20 at 15:05
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    I think this is likely to be closed as off-topic, but I think there is a slight misinterpretation here. The point Rodgers seems to me to be making is that mathematics ultimately rests on axioms, things that people agree on beforehand. What seems to us to be natural, as Yves claims, may not be natural to others whose perceptions are sufficiently different. If we were large enough to notice the curvature of the earth at every moment, perhaps we would not have Euclidean geometry, for instance. – Rylee Lyman Mar 16 '20 at 15:05
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    @RyleeLyman: though we live on a "flat" Earth, we discovered the non-Euclidean geometries, related to the independence of the axioms of the parallels. And I couldn't imagine little green mathematicians not creating a theory of sets such as ZFC. –  Mar 16 '20 at 15:21
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    @YvesDaoust I wonder if we lived on a very small sphere if "Euclidian geometry" would had been natural thought. And I am not convinced that green men would naturarily create ZFC, haven't many mathematicians looked for a better set theory for centuries? Maybe the little green man build a better model on the first try. – N. S. Mar 17 '20 at 03:34
  • It was a surprise for me that when, in past years, I was flipping through a book by Carl Sagan I found the section for appendix of his book. This is his book Cosmos, Random House (2002). I hope that this is the right reference, see the Wikipedia Cosmos (Carl Sagan book). Many times I wondered that if there exists an extraterrestial intelligence what about their books if are similar than Sagan's book and what subjects there are in the appendix that I evoke. Or if some open problem in mathematics is required to "travel" more faster than the light, since we know that "nothing" can to ... – user759001 May 13 '20 at 18:19

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Mathematics is absolute truth. But people tend to be wrong about how mathematics is absolute truth. They learn about numbers and shapes, and are eventually introduced to Euclidean geometry and its axioms, and they think "these axioms are universal truths, and everything follows from them logically".

And then, the horror sets in. They hear rumors about ... non-euclidean geometry. And they think "but that isn't real - it's just playing around". But then the physicists say "it is real". And their world falls about. There are no absolute truths! Everything is relative!

What they fail to realize is that they were misunderstanding the nature of mathematics all along. Mathematics isn't "real" and it never was. That is, it never deals with the physical world. It instead exists within a realm of thought only. Now, it has applications to the real world. It was invented (or discovered) exactly to describe things in the real world. But those applications are not part of mathematics itself. Whether or not those applications provide accurate or fallacious descriptions of the real world is not a mathematical question. Perfectly good mathematics has often been used for both. Those are questions for scientists and philosophers.

Yes, mathematical theories start with axioms. And yes, you can choose other axioms that disagree, and get another mathematical theory that is just as valid as the original. Which is correct? Both are! If you grew up only seeing and eating red apples, then one day went to a grocery store and saw green apples, should you recoil in horror and think your entire world has been turned upside down? Obviously, a better response would be "Neat! We get variety in apples!" The same is true here. Hyperbolic geometry is just a different variety of mathematics from Euclidean geometry. It is a whole new playground we can explore! Which is the "real geometry"? Well, what does "real geometry" even mean? The geometry of the physical world? That is a physics question, not a mathematical one.

So we abandon absolute truth? No. The axioms were never absolute. They are just definitions. They define the particular theory we are working in. But you are always free to define a different theory. What is absolute is not the axioms, but rather that the theorems in a theory follow from its axioms (based on its logic system - that can be redefined too). So I can't claim once and for all and ever more that through a point not on a line, there is exactly one other line parallel to the first. But I can say once and for all and ever more that in that system where there is only one parallel (amongst various other defining axioms), the interior angles of a triangle will add up to a straight line.

Paul Sinclair
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