Mathematical theories and axioms

In summary, a theory in mathematics is a set of formulas that are closed under logical deduction. A theory is considered axiomatic if there exists a finite or countable set of independent formulas from which all other formulas in the theory can be deduced. A formula is defined as a string of symbols specified by some grammar, and logical deduction is a set of rules that allow for the construction of new formulas from existing ones. While set theory itself is an axiomatic theory, the terms used in the definitions of theory and axiomatic must be carefully considered to avoid circular reasoning.
  • #1
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,877
422
Maybe this is a dumb question. I'm a bit tired right now. :smile:

What is a "theory" in mathematics, and what kind of statements can we call "axioms"?

To be more specific, is "group theory" a mathematical theory, and if yes, what are its axioms? Should I think of the definition of "group" as an axiom of the theory, or as "just a definition"?
 
Physics news on Phys.org
  • #2
I wouldn't say that the definition of a group is an axiom, but rather that each of the properties required in the definition of a group is an axiom.
 
  • #3
If you would like to know about theory in Math, look at Chapter 2 of Douglas Hofstadter's Godel, Escher, and Bach. It's a down-and-dirty quick explanation that ties it all together globally for you. He talks about theory and axioms and how they fit together, before he launches into his murky tirade about incompleteness/endless loops/canons...
 
  • #4
There isn't an essential difference between a definition and an axiom. And in practical terms, there isn't any essential difference between axioms and theorems either.
 
Last edited:
  • #5
Hurkyl said:
There isn't an essential difference between a definition and an axiom.

In metamath, definitions are just a kind of axiom. To me, though, the two are epistemologically different -- a definition, properly written, is always a conservative extension -- they never add expressive power. Ideally, the given axioms for a system each add expressive power, though proving this is hard in the light of incompleteness.
 
  • #6
CRGreathouse said:
In metamath, definitions are just a kind of axiom. To me, though, the two are epistemologically different -- a definition, properly written, is always a conservative extension -- they never add expressive power. Ideally, the given axioms for a system each add expressive power, though proving this is hard in the light of incompleteness.
I agree that there is pedagogical value in separating the ideas of axiom, definition, theorem, lemma, and so forth. I just worry that, sometimes, people read too much into the distinction.
 
  • #7
Thanks guys. Those answers are good enough for my purposes. (I'm just checking if I use those words the same as others).
 
  • #8
A "theory" is a set of formulas closed under logical deduction. A theory is "axiomatic" if there exists a finite (or countable) subset that are independent and from which every formula in the theory can be deduced. Think of it as a vector space with a "basis".
 
  • #9
Dragonfall said:
A "theory" is a set of formulas closed under logical deduction. A theory is "axiomatic" if there exists a finite (or countable) subset that are independent and from which every formula in the theory can be deduced. Think of it as a vector space with a "basis".
This is very interesting. It makes me want to learn more. :smile:

I have some follow-up questions:

a) Is there also a definition of what a "formula" is in this framework? Is it just a string of text?

b) Is there also a definition of what "logical deduction" is in this framework? Should we think of it as a set of rules that tell us how to construct a new formula from existing ones? (I guess "new" and "existing" are somewhat misleading terms here, since all the formulas of the theory already exist as members of the set). Wouldn't that make "logical deduction" a set of functions that take subsets of the theory (which is itself a set) to members of the theory?

c) Isn't set theory itself supposed to be an axiomatic theory? Then how can you use set-theoretic terms (e.g. subset) in the definitions of "theory" and "axiomatic"?
 
Last edited:
  • #10
(For the record, I'm not entirely convinced that the technical and informal uses of the word 'theory' are entirely the same)


a) Is there also a definition of what a "formula" is in this framework? Is it just a string of text?
Yes. You first define an alphabet to be a set of 'symbols'. Then, you define a string over that alphabet to be an ordered sequence of symbols. Finally, the 'language' of formulae is a set of strings specified by some grammar.


Is there also a definition of what "logical deduction" is in this framework? Should we think of it as a set of rules that tell us how to construct a new formula from existing ones?
Yes.

Wouldn't that make "logical deduction" a set of functions that take subsets of the theory (which is itself a set) to members of the theory?
Not quite; for any particular rule of deduction, only certain subsets of the language are in the domain of the rule.

Isn't set theory itself supposed to be an axiomatic theory? Then how can you use set-theoretic terms (e.g. subset) in the definitions of "theory" and "axiomatic"?
By being very careful about what you really mean. (And if you really want to talk about the ambient set theory, you invoke a metamathematical axiom that what mathematicians do is a model of formal logic)
 

Related to Mathematical theories and axioms

1. What are mathematical theories and axioms?

Mathematical theories are sets of mathematical principles and concepts that are used to explain and predict phenomena in the natural world. Axioms are statements that are taken as true without proof and serve as the basis for mathematical reasoning.

2. How are mathematical theories and axioms developed?

Mathematical theories and axioms are developed through a combination of observation, experimentation, and logical reasoning. Scientists and mathematicians use existing theories and axioms to make predictions and then test these predictions through experiments and observations. If the results match the predictions, the theory and axioms are considered valid.

3. What is the difference between a mathematical theory and a mathematical model?

A mathematical theory is a set of principles and concepts that explain a particular phenomenon, while a mathematical model is a representation of a system or process using mathematical equations. A theory can be used to develop a model, and the model can be used to make predictions based on the theory, but they are distinct concepts.

4. Can mathematical theories and axioms be proven?

No, mathematical theories and axioms cannot be proven. Axioms are accepted as true without proof, and theories are constantly evolving as new evidence and observations are discovered. However, theories and axioms can be supported and validated through experimentation and observation.

5. Why are mathematical theories and axioms important in science?

Mathematical theories and axioms play a crucial role in science because they provide a framework for understanding and predicting natural phenomena. They allow scientists to make accurate predictions and test their hypotheses, leading to a better understanding of the world around us and the development of new technologies.

Similar threads

Replies
72
Views
5K
Replies
2
Views
552
  • Quantum Interpretations and Foundations
Replies
10
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
625
  • Precalculus Mathematics Homework Help
Replies
13
Views
1K
  • General Math
Replies
6
Views
1K
Replies
8
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
  • General Math
Replies
1
Views
758
  • Quantum Interpretations and Foundations
Replies
15
Views
2K
Back
Top