Is it possible to calculate in physics with different sets of axioms?

In summary: Thinking that mathematics is based on axioms is an illusion. The axioms are merely a method of presenting the material and to put it on a rigorous basis.
  • #1
silenzer
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I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.
 
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  • #2
Can you clarify what you mean?

There is generally more than one set axioms that produces the same theory.
 
  • #3
Thanks for the reply. What I mean is that I'm debating with someone about whether mathematics is an invention or discovery, and he said that there is only one set of axioms in mathematics that produces results in physics. Is this true? So that, if we were to alter some of those mathematical axioms, the results would be incorrect.
 
  • #4
silenzer said:
Thanks for the reply. What I mean is that I'm debating with someone about whether mathematics is an invention or discovery, and he said that there is only one set of axioms in mathematics that produces results in physics. Is this true? So that, if we were to alter some of those mathematical axioms, the results would be incorrect.

Well, what he said is clearly false.

Do you know which axioms he's referring to?
 
  • #5
And I'll say it again. Thinking that mathematics is based on axioms is an illusion. The axioms are merely a method of presenting the material and to put it on a rigorous basis.

In actual mathematical research and discovery, axioms are rarely used. What happens is, we look at some basic examples and derive some general theory for that. Then we notice that our theory is very similar to some other theories that are developed. Finally, we abstract those theories to some more general theory. In order to present that abstract theory, we invent some axioms for them.

Everything we do in mathematics is in some way or another tied to nature. For example, the natural numbers are based on counting as we know it. The Peano axioms for the natural numbers were not put forward as some abstract entity. It's not that Peano said: "let's assume these axioms" and then suddenly found out that "hey, these are the natural numbers!". He made the axioms because he wanted to get the natural numbers. If they didn't give the natural numbers, then the axioms were wrong.

So mathematics is, in that sense, an experimental science. We see something that is interesting. And then we abstract this to a mathematical theory. I highly agree with Arnold when he says that: " Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. " See here for the interesting points of view of Arnold: http://pauli.uni-muenster.de/~munsteg/arnold.html

Anyway, given a certain theory, there are many possible axiom systems that will give you the theory. In fact, a mathematical theory consists of some physical examples that we want to abstract. If we can find axiom systems that encompass this, then we're done.

So, focussing on axioms is wrong. Axioms are a language, and not part of mathematics itself.
 
  • #6
silenzer said:
I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.

This is mostly a philosophy question, so that I don't believe it has a clear yes/no answer. There are many good books on it; some good ones are the "What is Mathematics", and many of the books by Ian Stewart. Antonio Damasio, a Neurologist ( or some type of brain scientist) put out a good book called "Where Mathematics Comes From". There is another good one by a UCBerkeley linguist whose name I can't remember now, but I'll think about it and get back to you. The good thing about Ian Stewart's book is that , on top of his being a great expositor, he has done research in just-about every area of Math one can think of.
 
  • #7
No math = magic.
 

1. Can different sets of axioms be used to calculate in physics?

Yes, different sets of axioms can be used in physics to calculate various phenomena. Axioms are fundamental principles or assumptions that serve as the basis for a mathematical or logical system. As long as the chosen set of axioms is consistent and complete, it can be used to make calculations in physics.

2. How do different sets of axioms affect calculations in physics?

Different sets of axioms can lead to different mathematical models and theories, which can result in different calculations and predictions. For example, using Euclidean geometry as opposed to non-Euclidean geometry can produce different results when calculating the curvature of space in general relativity.

3. Are some sets of axioms more accurate than others in physics?

It is not a matter of accuracy, but rather the relevance and applicability of the chosen set of axioms to the specific problem at hand. Some sets of axioms may be more suitable for certain phenomena, while others may be better for different situations. The validity and usefulness of an axiom set is determined by its ability to accurately describe and make predictions about the physical world.

4. Can different sets of axioms be used together in physics?

Yes, it is possible to combine different sets of axioms in physics. This can be seen in the development of new theories and models that build upon previously established axioms. For example, classical mechanics and quantum mechanics use different sets of axioms but can be combined to create the more comprehensive theory of quantum mechanics.

5. How do scientists choose which set of axioms to use in physics?

The choice of axioms depends on the specific problem being studied and the desired level of accuracy and complexity. Scientists often use well-established and tested sets of axioms that have been shown to accurately describe physical phenomena. However, in some cases, new sets of axioms may need to be developed to better explain or predict a particular phenomenon. Ultimately, the choice of axioms is based on their ability to accurately describe and explain the physical world.

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