Comments about "Axiom" in Wikipedia

This document contains comments about the article Axiom in Wikipedia
In the last paragraph I explain my own opinion.

Contents

Reflection

Introduction

The article starts with the following sentence.
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments
If it is true requires experiments.

1. Etymology

2 Historical development

2.1 Early Greeks

2.2 Modern development

A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom.
That seems as common sense.
In particular, the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.
Newton considers a clear separation between space and time.
In 1905, Newton's axioms were replaced by those of Albert Einstein's special relativity, and later on by those of general relativity.
Newton's Law is one side of a coin. SR and GR are a different coin with his own postulates.
The next paragraph has nothing to do with axioms or postulates

2.3 Other sciences

Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics.
Okay.
In particular, the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.
Newton's work is based on a clear separation between physical space and time.
In 1905, Newton's axioms were replaced by those of Albert Einstein's special relativity, and later on by those of general relativity.
Newton's axioms and the postulates of SR and GR have 'nothing' in common.
Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr, concerned the interpretation of quantum mechanics.
Quantum mechanics, which studies the behaviour of individual particles and clasical mechanics, which studies the behaviour of large objects in space have 'nothing' in common.
According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic.
The true question is the physical world (the evolution of the processes under investigation) probabilistic or deterministic? A more fundamental question is: can we (humans) predict the future? The answer is no. We can only predict the future with a certain accuracy.
The point is when you go to more detail and come at particle level, to repeat a process becomes more and more difficult and probabilistic. In that sense the earth is not round which influences the predictions of its future trajectory.
As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments.
This is a completely incorrect point of view. In any discussion it should be understood that all parties involved, explain what they mean in clear language, and what the constrains (assumptions, boundaries) are. At the same time all parties should also be able to explain the work and opinions of all the other parties.
Regardless, the role of axioms in mathematics and in the above-mentioned sciences is different. In mathematics one neither "proves" nor "disproves" an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments always makes sense, since a falsified physical theory needs modification.
The point is that it is tricky to compare axioms in mathematics with the axioms in physics.
The axioms and logic used in science are very rigid. No experiments are involved to test the results.
In physics axioms and assumptions are very important. Mathematics is a tool used to predict the results of the experiments. However this interplay between these two fields, always introduces uncertainties, which are of importance for the final results. A doctor never knows the exact condition of his patients, which can be important if he proposes a cure.

3. Mathematical logic

3.1 Logical axioms

3.1.1 Examples

3.1.1.1 Propositional logic

3.1.1.2 First-order logic

3.2 Non-logical axioms

Another name for a non-logical axiom is postulate.
Is this the mainstream opinion?

3.2.1 Examples

3.2.1.1 Arithmetic

3.2.1.2 Euclidean geometry

3.2.1.3 Real analysis

3.3 Role in mathematical logic

3.3.1 Deductive systems and completeness

3.4 Further discussion

4. See also

Following is a list with "Comments in Wikipedia" about related subjects

Reflection 1 - axioms versus postulates

In this document the concepts axioms and postulates are considered identical (but not completely). IMO it make more sense to use the word axioms for mathematical purposes and the word postulates for physical processes. Axioms are more bassic.

The problem with the word postulate is that it resembles the word law.

Reflection 2 - critical evaluation postulates of SR

The first postulate of SR is about the speed of light. The postulate claims that the speed of light in an inertial frame in all directions is the same. To write it different: the speed of light of light in both directions from an observer A at rest in an inertial frame is the same.
If that is the case than if observer A emits a flash of light in al directions she will stay at the center of this sphere which propagates in all directions with the speed c.
Now suppose observer A starts to move towards the right, will she stay at the center of this sphere?
I doubt this.
Was she ever at the center of this sphere?
When it is true that when she moves away, she moves out of the center, than most probably she was never in the center.
Does that mean that the flash of light does not propagate in a sphere? No, it does not. The flash of light propagates in a sphere but completely independent of the speed of the source.
Generally speaking you can have two sources which simultaneous emit a flash of light. From a physical point of view there is no difference between the two (except frequency) and neither of the two sources will stay at the center of the sphere.

What this means from a physical point of view that the sphere propagates with the speed c but that the speed of the source is not known.

The second postulate of SR claims that all the physical laws in each inertial reference frame are identical.
Suppose that that is the case, does that in any way decides what the laws of physics are? I doubt this.
When you want to unravel the laws of physics you should start from one reference frame and make all the observations in one and the same frame and see which are the parameters that influence each process under consideration to learn how it works and which are the reactions in time.

Reflection 3

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Created: 3 November 2017
Modified: 16 June 2019

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