Do We Assume Logic for Mathematics?

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In summary, mathematics is a self-contained system that relies on a set of assumed axioms to construct theories and proofs. These axioms must be logically consistent in order for mathematics to function properly. However, the consistency of these axioms cannot be proven within the system itself. While most mathematicians assume that the commonly used ZFC set of axioms is consistent, it has been proven to be incomplete and its consistency can never be proven. This raises questions about the nature of logic and whether it can be defined by its own rules.
  • #1
V0ODO0CH1LD
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do we assume "logic"?

Mathematics has no foundations on reality, it stands on its own. But to construct it we have to assume a number of axioms, like if I wanted to create a "science" where the only rules are all circles are red and all squares are blue, then from that we can build theories and prove them from the assumed axioms, like a particular square has to be blue because all squares are blue, right?

However, if the defining rules of my "science" are contradictory, like all circles are red and all circles are blue, then my "science" is "ill defined".

But what indicates that? If the cornerstones of mathematics have to be "logical", not in the sense of mathematical logic or propositional logic per se but in the sense that they have to be "well defined", does that mean that we are assuming a set of implicit primordial rules that the rest of mathematics have to abide to?

If even the most basic laws of mathematics have to be "logical", does that mean that we are assuming a set of rules that dictate whether something is logical or not? And could these rules be "logical" if they define logic? In other words, can they abide by their own rules? Is that even possible?
 
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  • #2
The short answer to your question is ... yes, most working mathematicians do their work with under the assumption that ZFC (the "math world" in which 99% of them work) is consistent (there are no contradictions) and complete (everything that is true is also provable and vice versa).

You should look into topics such as Godel's Incompleteness Theorem, metamathematics, and philosophy of math for the beginnings of the long answer to your concerns.
 
  • #3
However, just as we can work in many different sets of axioms, getting different mathematical "stuctures" so we can work in different "logics", ZFC or not. And just as we have to specify what axioms we are using, we have to specify what logic.
 
  • #4
gopher_p said:
The short answer to your question is ... yes, most working mathematicians do their work with under the assumption that ZFC (the "math world" in which 99% of them work) is consistent (there are no contradictions) and complete (everything that is true is also provable and vice versa).

ZFC has been proven to be incomplete. The continuum hypothesis for example cannot be proven.

Most mathematicians do believe that ZFC is consistent. But this can never be proven. In fact, whether ZFC is consistent is totally irrelevant to most. If it were inconsistent, we would find a new axiom system and formulate all of our mathematics in there. Most mathematicians wouldn't even notice anything changed.
 
  • #5
R136a1 said:
ZFC has been proven to be incomplete.

This is assuming Con(ZFC) of course :tongue:

Most mathematicians do believe that ZFC is consistent. But this can never be proven.

Assuming Con(ZFC) then yes ZFC cannot prove its own consistency. This is technically different from not being provable altogether since you could hypothetically move to another set theoretic universe where you can prove consistency. Results of this nature are actually of some interest. For example PA is unable to prove its own consistency yet we have results like this: http://en.wikipedia.org/wiki/Gentzen's_consistency_proof
 

1. What is the relationship between logic and mathematics?

The relationship between logic and mathematics is that logic provides the foundation and framework for mathematics. Logic is a system of reasoning and rules for determining the validity of arguments, while mathematics is the study of quantity, structure, and change. Mathematics relies on logic to ensure that its arguments and conclusions are sound and valid.

2. Can mathematics exist without logic?

No, mathematics cannot exist without logic. Logic is essential for constructing and validating mathematical arguments and proofs. Without logic, mathematics would lack coherence and consistency, making it impossible to make accurate and reliable mathematical statements and conclusions.

3. Is logic assumed in all branches of mathematics?

Yes, logic is assumed in all branches of mathematics. Whether it is algebra, geometry, calculus, or statistics, all mathematical concepts and principles are built upon logical reasoning. Without logic, it would be impossible to make meaningful and valid mathematical statements and proofs.

4. How does the use of logic affect the development of mathematics?

The use of logic is crucial for the development of mathematics. It allows mathematicians to make logical deductions and conclusions based on established rules and principles. This helps to build a strong foundation for further exploration and development of new mathematical concepts and ideas.

5. Can there be different types of logic used in mathematics?

Yes, there can be different types of logic used in mathematics. While classical logic is the most commonly used in mathematics, there are other types of logic, such as modal logic and intuitionistic logic, that can be applied to specific areas of mathematics. The choice of logic often depends on the context and the type of mathematical problem being studied.

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