Does what happened with the emergence of non-Euclidean geometries deny the existence of axioms in the innate sense?

Question

I read this text a while ago (in Arabic):

Supporters of contemporary mathematics believe that mathematics results are relative in certainty and not fixed, because the emergence of the axiom system destroyed the principle of a priori, so what is known as the crisis of mathematical certainty emerged, and all mathematical issues became mere debatable priorities, which is what made mathematics able to keep pace with the progress of science. The crisis of mathematical certainty began with Leibnitz, who saw that if the mathematical structure is based on a set of principles that are self-evident and do not require proof, then why should we doubt the Euclidean system if its principles are not self-evident in the innate sense of the word, but rather are merely hypothetical issues? This proposal was met with In the 20th century AD, contemporary mathematical scientists noticed that there is no shame or embarrassment in reconsidering mathematical principles, and they rejected the existence of fixed principles. Rather, they are merely starting points from which the mathematician can establish whatever he wants. On this basis, the Axiomian system appeared and destroyed the idea of ​​a priori with multiple mathematical systems. The Russian mathematician Lobachevsky and the German mathematician Riemann. The first assumed a concave shape for the place and from that he concluded that the sum of the angles of the triangle is less than 180°. Riemann assumed that the place has a spherical shape and concluded that the sum of the angles of the triangle is greater than 180°. The multiplicity of mathematical systems according to them is evidence of the relativity of this. Science because multiplicity indicates relativity. Boligan says: [The abundance of systems in geometry is evidence that mathematics does not contain absolute truths].

I want to say that there is an alien philosophy that has been pasted into the interpretation of what happened to reduce the value of religion and certainty in anything; They did not prove that the axiom that had been accepted for more than two thousand years was wrong (the axiom of parallelism). They changed the definition they started from and gave it to the same word. It is natural for them to reach worlds different from the Euclidean world. Vlobachowski and Riemann did not deal with the straight line with the same concept as Euclid, but rather they dealt with another, broader being. From Euclid's line it can work with different worlds consistently; This does not mean that Euclid's axiom is wrong, Yes, Euclid may not have defined the line well, but through his implicit use of the rectum we can understand exactly what he meant.

Can the futility of innate intuition really be deduced from the emergence of non-Euclidean geometries? Or is it just an interpretation of things from an atheistic, materialistic, or agnostic perspective?

Answer 1

The suggestions from the long paragraph quoted in the question seems to put everything on its head.

Mathematics is not "relative" because the "axioms are not fixed".

Axioms are a required starting point for any tree of logical proofs, not a liability. That's the way logic works - if you have absolutely nothing to go off on, then you cannot do logic.

A choice of axioms is not something that needs to be fixed in time forever. The important part about a set of axioms is that it is valid and beneficial to the topic you want to study.

This allows plenty of good things:

• You can pick-and-choose your axioms to make your proofs easier or possible. As long as the choice of axioms means that whatever you are developing has the utility you want it to have, that is perfectly fine, and not "cheating" in any respect.
• As a corollary, it allows and encourages you to pick a minimal set of axioms for clarity and generality.
• What is an axiom in one particular model could be a regular proof in a different one (with different axioms). This way, you can take over all the insights from previous research by "proving" what has been axioms in the past.
• The opposite is of course true. If you manage to infer a contradiction from your set of axioms, you know that those particular axioms are not compatible, and that you need to remove or otherwise modify them.
• And finally, as the OP alludes, modifying axioms over time does in fact allow maths to keep up with the times, and integrate new developments in science (or, more likely, maths itself).

Mathematics was not in a foundational crisis because axioms changed, but some common axioms were changed because of the crisis (to try and solve it or at least make it so we can live with it). The crisis (in the times of Cantor) was because the particular axioms that had been in use past then (in set theory) turned out to lead to inconsistent or otherwise unsatisfactory results when building the rest of mathematics on top of them. It was the great work of Gödel who showed without a doubt that it is principally impossible to have a logic that is both internally consistent and powerful. Now that we know this, and have accepted it, we can put this "crisis" to rest - people know how to work around these problems, and are not surprised when they crop up.

Answer 2

1. It seems not helpful to connect the axiomatizing of mathematical theories with ideological positions “from an atheistic, materialistic, or agnostic perspective” – as the last sentence in your post does.

   Mathematics and ideology are separate domains. Mixing-up the two only promotes defaming mathematicians who do not belong to the own camp.

2. In the 19th century mathematicians made the discovery that the parallel postulate is independent from the other Euclidean axioms of geometry. This insight prompted the development of non-Euclidean geometries, which later were fundamental for the General Theory of Relativity.

3. From the view point of epistemology the discovery of non-Euclidean geometry showed that “intuition” can be a prejudice, which possibly leads to an unjustified restriction of thoughts. Intution can be a good heuristic, but it must not narrow-down the search space for ideas.

   Hence my answer to your title question: Yes, the emergence of non-Euclidean geometries shows that intuition cannot be the final word, neither when choosing the axioms of a mathematical theory nor when proving a mathematical theorem.

Aside: I do not understand what “innate intuition” means in comparison to just “intution”.

Answer 3

The claim that the emergence of non-Euclidean geometries demonstrates the "futility of innate intuition" can be interpreted in several ways, depending on what is meant by "innate intuition". If innate intuition refers to an inborn understanding or insight into the nature of reality, then the discovery of non-Euclidean geometries could be seen as challenging the idea that our intuitive grasp of space and geometry reflects the true nature of the physical universe. It can be argued that our everyday experiences and intuitions are rooted in a Euclidean framework, whereas the universe may not conform to this framework at all scales or in all contexts (as suggested by General Relativity, which models gravity using the mathematics of non-Euclidean geometry).

However, interpreting these developments from an atheistic, materialistic, or agnostic perspective is not necessarily the only or inevitable conclusion. The interpretation largely depends on one's philosophical or metaphysical outlook. As Jo Wehler has pointed out, the emergence of non-Euclidean geometries more probably reflects the fact that our [innate] intuitions are by no means a secure source of knowledge, nor do they provide a firm foundation on which to build abstract thinking schemes that have all the generlity possible to represent arbitrarily complicated data.

Answer 4

I will focus on the question you have asked in your comments in response to other answers, namely whether Euclidian geometry is 'wrong'. There are two distinct aspects to the question. The first is that geometries can be considered as mathematical structures, the properties of which depend on their axioms, so in that sense Euclidian geometry is certainly not wrong. The second is whether or not a particular geometry reflects the nature of reality around us. In that sense, 3-D Euclidian geometry is a very accurate model of space on an everyday scale. I used Euclidian geometry to calculate the length and shape of the rafters in my house, and any errors arising from that choice were dwarfed by my errors in making measurements and cuts. However, 4-D Euclidian geometry is not a good model of spacetime, so it is 'wrong' in that sense.

Some human intuitions seem useful, no doubt because we would not have survived as a race without them, but our 'innate intuitions' have developed through evolution to suit the environment in which we evolved and to complement our other capabilities, such as our senses of sight, touch etc. Since those senses are naturally very limited (we can't see things smaller than a certain size, for example), it should not be surprising that our intuitions fail when we try to extend them beyond our limited natural perspective on the world.

Answer 5

To add another example or two:

Once you invent subtraction on the number line, you are required to modify the line to include a new portion that stretches to negative infinity- and also the origin at zero. With zero in hand, the field expands.

Once you invent division, you must further expand the number line to include all the rationals that lie between any two integers on the line.

Once you invent square roots, then you must expand the number line to include the irrationals that lie between any pair of rational numbers.

Once you invent negative square roots, you must expand the number line to include a second number line at right angles to the original on which the imaginaries exist.

The concepts of zero, negative numbers, rational numbers, irrational numbers and complex numbers were at one time all considered counterintuitive and nonsensical- until they weren't. And thus the field expands.