What Mathematical Logic Says about the Foundations of Mathematics
📝 Abstract
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can “explain”. For example, let’s consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, to present, and to clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are interesting also because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?
💡 Analysis
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can “explain”. For example, let’s consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, to present, and to clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are interesting also because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?
📄 Content
CHAPTER THREE
WHAT MATHEMATICAL LOGIC SAYS ABOUT
THE FOUNDATIONS OF MATHEMATICS
CLAUDIO BERNARDI
SUMMARY My purpose is to examine some concepts of mathematical logic studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can “explain”. For example, let us consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, present, and clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are also interesting because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?
KEYWORDS Mathematical logic, foundations of mathematics, axiomatic systems, proofs, definitions, mathematical education.
- Mathematical logic vs. the foundations of mathematics
There is no doubt that research in mathematical logic can contribute to the study of the foundations of mathematics. For instance, mathematical logic provides answers (both complete and partial) to the following questions:
Given a precisely stated conjecture, can we be sure that eventually a good enough mathematician will be able to prove or disprove it?
Can all mathematics be formalized? Chapter Three
42
Is there a “right” set of axioms for arithmetic, or for mathematical analysis? is there a proof (in some fixed theory) for any statement of arithmetic which is true in N?
Can we prove the consistency of standard mathematical theories? and what does it mean to prove consistency?
By adding a new axiom to a theory, we find new theorems; but can we also expect to find shorter proofs for old theorems?
Will we ever construct a computer that will be capable of answering all mathematical problems?
Is any function from N to N computable by an appropriate computer? if not, how can we describe computable functions?
If we know that a computation ends, can we estimate the time necessary to complete the computation?
Is it true that, if a “short” statement is a theorem, then there is a short proof for it? The list could be much longer. In some cases (as in the first question) the answer given by logic contradicts the expectations of a working mathematician, while in other cases (as in the last question) the answer confirms that expectation. However, it is not true that the general purpose of mathematical logic is to clarify the foundations of mathematics. First of all, for the past few decades, much research in logic has been of mainly technical value and does not deal directly with the foundation of mathematics. Perhaps in the nineteenth century, logic was regarded as a way to guarantee the certainty of mathematics. But nowadays we do not expect that much: it seems naïve, and perhaps even futile, to hope for a definitive, proven certainty of mathematics. Let us start from the beginning. Mathematical logic provides us with a precise definition of a proof and suggests rigorous methods and procedures for developing mathematical theories. But these are just the initial steps of mathematical logic: if logic consisted only in giving detailed definitions of proofs and theories, it would not be of great scientific importance. While succeeding in formalizing statements and arguments is interesting, the historical and cultural importance of proof theory, model theory, and recursion theory strongly depends on the results achieved in these areas (for example, on the answers given to the previous questions). In other words, mathematical logic is a way of organizing mathematics and solving paradoxes; but I find that logic is interesting also because its organization of mathematics provides significant results. In fact, any theory grows if and when results are found.
What Mathematical Logic Says about the Foundations of Mathematics
43 So, we can distinguish between two kinds of logical results which can be useful in the study of foundations and, more generally, to working mathematicians. On the one hand, mathematical logic provides explicit rules that mathematicians habitually use (often without being fully aware of it), inserting them into a clear and consistent framework; in this way more complex situations can be tackled. For instance in logic:
it is explained what a proof by contradiction, or a counterexample, is; it is not impossible for a mathematician, who in his work usually gives proofs by contradiction and counterexamples, to be unable to give clear answers to explain these totally
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