Is Mathematics a Subset of Logic?

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Joseph47
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Is Mathematics a Subset of Logic?

Post by Joseph47 » November 10th, 2018, 12:08 pm

I have done a little searching on the Web concerning logic versus mathematics, and it seems that most people whose ideas about logic and mathematics that I've encountered conclude that logic is a branch of mathematics. A mathematician explicitly stated that in a book I read. But is that correct?

Is not logic a set of principles that undergirds all language, all reasoning, and all relationships between objects and entities? If this is the case, then I ask if mathematics undergirds all reasoning, all language, and all relationships between all realities -- including relationships between thoughts and ideas. It seems evident to me that formal symbolic language is but a tiny subset of logic and that mathematics is also a subset of the "universal" logic that undergirds all reality.

I wish for replies to my question: Is mathematics a subset of logic? If not, how can the claim that all logical reasoning is a branch of mathematics be defended? By the way, I just looked up "logic" in my The American Heritage Dictionary and found several definitions. I want to list two of the definitions. Definition 2c.: "The formal guiding principles of a discipline, school, or science." Def. 4: "The relationship between elements and between an element and the whole in a set of objects, individuals, principles or events...."

How could mathematics be properly claimed to describe or define every set of elements and relationships between elements, including moral principles, an intensity of desire, etc.? It appears to me evident that mathematics cannot subsume all logical principles or all of logical reasoning. And is not logic included in every principle of logical reasoning?

Does anyone else have thoughts on this issue? I've given much thought to it over several decades, and I can hardly believe that it is proper to declare that logic (including all the definitions of logic) is a subset of mathematics. As I see it, the converse is the case.

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ktz
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Re: Is Mathematics a Subset of Logic?

Post by ktz » November 10th, 2018, 5:36 pm

Sounds like a semantic issue. It's important to distinguish between formal logic and propositional calculus, and just run of the mill informal usages of the term logic. Mathematics gives rigorous definitions to things within the scope of its field of study -- for example, a "group" in mathematics has a very distinct definition compared to informal usage of the term, allowing for certain kinds of verifiable mathematical proofs and theorems to be made. When a mathematician says that logic is a branch of mathematics, what he is saying, per wikipedia, is that there are groups of researchers pursuing two distinct areas of research: the application of the techniques of formal logic to mathematics and mathematical reasoning, as well as in the other direction, the application of mathematical techniques to the representation and analysis of formal logic. Logic can still exist in informal forms outside of mathematics, or even other formal forms like in the context of philosophy or computational logic in computer science.

Mathematics is basically just the scientific study of number, quantity, and space. Logic is one area of study among many, and provides a set of tools to build upon the basic axiomatic foundations of mathematics. I don't think math subsumes the entire concept of logic nor the converse -- trying to make the case that mathematics is a subset of logic, to me that sounds like saying carpentry is a subset of tools to make wooden stuff. There are somewhat rigorous semantic meanings within these contexts, so I feel like it's important not to conflate informal ideas about them when reasoning about them.

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Re: Is Mathematics a Subset of Logic?

Post by Joseph47 » November 12th, 2018, 8:49 pm

ktz wrote:
November 10th, 2018, 5:36 pm
Sounds like a semantic issue. It's important to distinguish between formal logic and propositional calculus, and just run of the mill informal usages of the term logic. Mathematics gives rigorous definitions to things within the scope of its field of study -- for example, a "group" in mathematics has a very distinct definition compared to informal usage of the term, allowing for certain kinds of verifiable mathematical proofs and theorems to be made. When a mathematician says that logic is a branch of mathematics, what he is saying, per wikipedia, is that there are groups of researchers pursuing two distinct areas of research: the application of the techniques of formal logic to mathematics and mathematical reasoning, as well as in the other direction, the application of mathematical techniques to the representation and analysis of formal logic. Logic can still exist in informal forms outside of mathematics, or even other formal forms like in the context of philosophy or computational logic in computer science.

Mathematics is basically just the scientific study of number, quantity, and space. Logic is one area of study among many, and provides a set of tools to build upon the basic axiomatic foundations of mathematics. I don't think math subsumes the entire concept of logic nor the converse -- trying to make the case that mathematics is a subset of logic, to me that sounds like saying carpentry is a subset of tools to make wooden stuff. There are somewhat rigorous semantic meanings within these contexts, so I feel like it's important not to conflate informal ideas about them when reasoning about them.
You might be right that it's mainly a matter of semantics. After all, words only mean what we define them to mean. However, when a mathematician writes in a mathematics book that we arrive at a criterion for the number concept "...Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiphased necessity that we call mathematics...", I believe that there's more than semantics involved here. I believe that this mathematician, Tobias Dantzig, was intending to say that, whatever mathematics and logic are, logic is only a phase of mathematics. This seems, to me, terribly wrong-headed. Mathematics, as I see it, arises from the logical necessities that inhere in specifying relationships within the space-time continuum. My quote above was from Dantzig's book, Number: The Language of Science. It's a very good book, but I had some disagreements with certain of the philosophical claims made by the author. Of course, he was a mathematician, not a philosopher.

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Re: Is Mathematics a Subset of Logic?

Post by ktz » November 13th, 2018, 9:24 am

Oh, well the book you are citing is originally from 1930, during the pre-Godelian time when the ideas of Frege and Russell and Principia Mathematica held wide influence, and logicism was a popular conception. This is not nearly as popular of an idea today, you can read about some of the problems facing logicism/neologicism at SEP: https://plato.stanford.edu/entries/logi ... mProForLog

It's easy to take Dantzig's writings with a grain of salt when he himself writes at the beginning of the chapter you reference, "...by selecting reality as the theme of this concluding chapter, I am encroaching on a field foreign to my training... My interest lies exclusively in the position which the science of number occupies with respect to the general body of human knowledge." And as long as we're comfortable in the realm of semantics, saying that logic has no existence independent of mathematics may simply be saying that they are not distinct enough for logic to be used as a criterion for the connection to reality. It may not necessarily be implying that one subsumes the other -- Dantzig may be using the term "phase" metaphorically in the chemical sense like gas or liquid.

I also want to contest your use of the term logic as "set of principles that undergirds all language, all reasoning, and all relationships between objects and entities". This is an extremely broad usage of the term that would seemingly seek to include all of linguistics, philosophy, and physics. Consider for example the usages of the term "illogical". There is reasoning that occurs within the realm of intuition and "gut feelings" are not strictly logical, in that they are not necessarily bound to known facts and logical principles. There are events that occur that can also be characterized as illogical outcomes, because an analysis of the known data would not have predicted their occurence. Thus we can start to draw a boundary, as logic generally refers to mathematical operators that allow us to draw inferences based on known information. Logic as an abstraction operates within the realm of known quantities, and the universe contains plenty of unknowns about which logical reasoning is not yet possible, requiring first mathematical or other observational characterizations so that we can understand them well enough to begin to apply logical tools to their existence.

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Re: Is Mathematics a Subset of Logic?

Post by Joseph47 » November 14th, 2018, 11:14 am

ktz wrote:
November 13th, 2018, 9:24 am
Oh, well the book you are citing is originally from 1930, during the pre-Godelian time when the ideas of Frege and Russell and Principia Mathematica held wide influence, and logicism was a popular conception. This is not nearly as popular of an idea today, you can read about some of the problems facing logicism/neologicism at SEP: https://plato.stanford.edu/entries/logi ... mProForLog

It's easy to take Dantzig's writings with a grain of salt when he himself writes at the beginning of the chapter you reference, "...by selecting reality as the theme of this concluding chapter, I am encroaching on a field foreign to my training... My interest lies exclusively in the position which the science of number occupies with respect to the general body of human knowledge." And as long as we're comfortable in the realm of semantics, saying that logic has no existence independent of mathematics may simply be saying that they are not distinct enough for logic to be used as a criterion for the connection to reality. It may not necessarily be implying that one subsumes the other -- Dantzig may be using the term "phase" metaphorically in the chemical sense like gas or liquid.

I also want to contest your use of the term logic as "set of principles that undergirds all language, all reasoning, and all relationships between objects and entities". This is an extremely broad usage of the term that would seemingly seek to include all of linguistics, philosophy, and physics. Consider for example the usages of the term "illogical". There is reasoning that occurs within the realm of intuition and "gut feelings" are not strictly logical, in that they are not necessarily bound to known facts and logical principles. There are events that occur that can also be characterized as illogical outcomes, because an analysis of the known data would not have predicted their occurence. Thus we can start to draw a boundary, as logic generally refers to mathematical operators that allow us to draw inferences based on known information. Logic as an abstraction operates within the realm of known quantities, and the universe contains plenty of unknowns about which logical reasoning is not yet possible, requiring first mathematical or other observational characterizations so that we can understand them well enough to begin to apply logical tools to their existence.

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