What separates mathematics from logic? Can "mathematical" operations be applied to logical systems?

Question

In my 'Introduction to Logic' class, my professor told us that half of the class will be based on "mathematical" operations withing logic. After looking through the textbook, I realized that he meant things like the predicate calculus and propositional logic. I know that he probably just meant that these logic topics are "mathematical" in that they require certain symbol interpretation and manipulation, but it got me thinking about the essence of a mathematical system vs. a logical one. Set Theory, and similar foundational theories (such as Proof Theory), are based on axiomatic systems that are built upon the rules of logic. Of course, it depends on the textbook one is using to learn the mathematical system, as different logical symbols and relationships may be referenced in the system. For example, Kunen's book on Set Theory and foundations uses the first-order predicate calculus (if I remember correctly), so the \forall symbol is defined based on relationships of other symbols. Other textbooks may strictly define the symbol as its own symbol.

However, would it be valid to define logical operations as mathematical ones? The logical "or" symbol can arguably be a mathematical symbol (union in Set Theory). But Set Theory is itself based on these logical rules, so is it then not recursive to say that logic is based on Set Theory which is based on Logic? There are also things like Number Theory or Abstract Algebra, which are not necessarily based on any logical rules (unless one formalizes the Peano axioms and such). Does it still then follow that mathematics is built on logic? And if so, are the logical operations upon which it is built considered "mathematical" operations? What even qualifies operations as being "mathematical"?

At the core of my questions is the concept of mathematical operations and whether or not such operations are valid within logical systems. I also wonder if mathematical operations simply depend on interpretation and meaning rather than something innate, meaning any mathematical operation could be a logical one depending on how we define it.

This is only an introduction for me so please excuse my ignorance.

Answer 1

Short Answer

Both disciplines use symbols in a truth-centered, rule-based meaningful way, but mathematics is built on logic and is more contextual and covers topics such as known and unknown quantities, length, area, volume, direction and position, and shapes and their transformations. For instance, even simple arithmetic tends to be "built" on logical theorems.

Long Answer

This is no small ask. What you ask after is 'what is the nature of the intersection of logic and mathematics?' Perhaps the most famous person to ask this question is Gottlob Frege. This is a question related to both the philosophy of mathematics and the foundations of mathematics. Is this another demarcation problem like that which lies at the heart of differentiating and generalizing scientific methods from pseudoscience and each other?

Traditionally, formal and informal logics have been attempts to understand how people reason generally and obtain to the art of argumentation. That patterns of thought can be symbolized and turned into formal systems of symbols which are an extension of formal languages is true by definition. See Wikipedia's wonderful diagram of the syntactic divisions of a formal system in the article 'formal languages' to see how sequences of symbols when well-formed can be considered theorems by the application of truth values. To reiterate, when a sequence of symbols is considered acceptable (one might describe such values using BNF) and those acceptable strings are true, then one has an axiom or theorem. An axiom is presumed true, and a theorem is shown to be logically equivalent to the axioms. That is the essence of a formal system. So, to understand the relationship between logic and mathematics, one needs to see that a formal system is composed of four parts. From WP:

1. A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
2. A grammar consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
3. A set of axioms, or axiom schemata, consisting of well-formed formulas.
4. A set of inference rules. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.

The first two points are the essence of a formal language, and the last two added are the criteria for a formal system. Both logic and mathematics can be done according to their formal systems. There is not a limit to logics. Boole had his algebra, and there's FOPC. Modal logic, infinitary logic, and intuitionistic logic are more advanced logic that math majors tend not to study.

All of these logics have an essence. They take input statements which includes variables and relations, transform them with logical operations, and output statements. Where mathematics differs is that it tends to have more semantic information attached to it. Mathematics considers more broadly notions that apply to shapes, known and unknown quantities, natural language, and direction. That is logic is necessary to do math, but it is not sufficient. Some examples:

In logic, one sees logical equivalence (<-->, IFF). But in mathematics, the notion of identity is much broader. In arithmetic and algebra, it is definition (let some number be equal), equality (given operations sum total, it turns out to be equal), and identity (by substiution, the formulas are equivalent); in geometry it might be similarity (same shape, different quantities) or congruence (same shape, same sizes). ALL of those instances are examples of logical identity being used in a specific context be it assignment or comparison.

Note that as one can't escape logic in arithmetic, one cannot also escape arithmetic in logic. It has been long recognized by Quine and others that existential operator is arithmetic in nature. 'There exists a unique x such that' (∃!x:) is logic jargon for 'some set S there is exactly one element x (|S|=1:x∈S)'. Additionally, set theory can be used to define arithmetic operators such that addition can be defined in terms of union, subtraction in terms of set difference, etc. But whether you're ordering numbers on the number line, or determining if a set is a proper subset of another, you still need to use the fundamental logic operators to have statements and evaluate their truth relations. This is why whether dealing with model theory (defined often as 'universal algebra + logic') or proof theory both the model operator (⊨) and the proof operator (⊢) are both just contextual examples of logical implication (→) (in this case a metalanguage to describe the logical relationship among logical statements in a formal system).

Answer 2

Mathematical logic is a species of symbolic logic, itself a species of formal logic, which started essentially with Aristotle's syllogistic 2,500 years ago.

Formal logic has always been understood by logicians as an attempt to represent or model human deductive thought. Mathematical logic, too, was initially an attempt to model what Boole called the "laws of thought", i.e. human logic, using a symbolic notation rather than verbal arguments used in the Aristotelian tradition.

Thus, strictly speaking, the only logic we know is the logic of human deductive thought, best understood as a property or a capacity of the human mind, or of the human brain.

Symbolic logic is obviously a part of mathematics. Like any mathematical discipline, it is rigorously logical. However, as the label suggests, it is also supposed to be a species of formal logic, i.e. a way to model the logic of human deductive thinking.

Mathematical logic is of course mathematics. However, George Boole's explicit objective of producing a mathematical model of the laws of thought got largely forgotten starting with at least Bertrand Russell, so that now, whether mathematical logic is understood by mathematicians themselves as a model of human logic depends in fact on each mathematician.

Essentially, mathematical logic is a symbolic system which is mathematical but not a model of human logic. It is certainly not demonstrated that it is. It is also apparent that mathematical logic has only a minimal influence on how mathematicians outside mathematical logic really prove theorems.

Apparently, mathematicians essentially work as they did prior to the introduction of mathematical logic. This is certainly what any mathematical textbook outside mathematical logic suggests. Proofs of mathematical theorems are the same sort of semi-formal proofs done before mathematical logic, and never formal proofs as done in mathematical logic.

So, according to the most reasonable of interpretations, mathematical logic is not a model of the logic of human deductive thinking, and therefore, strictly speaking, though logical, not formal logic at all. It is essentially a mathematical discipline somewhat inspired by human logic.

So the relation between logic and mathematics is only that, as Aristotle could have quipped, all mathematicians are human beings, and all human beings are logical, therefore all mathematicians are logical.

Answer 3

As you can see from other answers here, philosophers tend to see mathematics as a branch of logic - a very big one, of course, but child to the general parent of logical and reasoned discourse none the less.

Mathematicians tend to see it the other way round. They will inform you earnestly that all forms of reasoned discourse are simply applications of the pure mathematics of some particular logical system. Formal logic effectively began with Euclid's axiomatization of geometry, and that is firmly in the realm of mathematics.

My own feeling is that when Hell freezes over, the Four Horsemen of the Apocalypse will still be arguing the toss.

Answer 4

My argument starts from an explicitly Kantian set of premises, so 😨 now, let's say we have intuitive and discursive knowledge. Waiving forms/formal intuition of space or time in particular, let's just say, but what is our knowledge of the difference? Do we intuit that there is discursion and intuition, or do we know this discursively? Or both?

If both, and to advert to at least the phrases of faculty psychology, then is there a form of knowledge that is not just given with both as both, but by some third "faculty"? What is this quasi-intuitive, quasi-discursive cognition...?

But Kant explains the difference as: intuition is of particulars, discursion operates on generality first. So our possible third quasi-faculty has to do with this. Numerical identity and haecceities become the subject. What knowledge do we have of differentiation as such? It is enough for something to be unique if it is indexed uniquely. What is this pure indexicality? But every number is differentiated unto itself. Either it has a finite number of digits, or an infinite number. Real numbers have aleph-0 many digits. In a sense, then, there must be numbers with aleph-1 many digits, aleph-2 many digits, and so on. And this is besides the alephs themselves (and it would be intangible, to speak of numbers with as many digits as there are elements of a measurable cardinal).

Now if you subscribe to a strong enough game-formalism about math and logic either way, you might end up thinking of this demi-intuition as like our knowledge of rules of games, including language games. But it would be better described as knowledge of a formal game itself, transcendental (encoded into the form of knowledge per se for us) for games at least, and then neither sense nor reference, but again indexicality, for itself. Numbers are possible scores in this game, if you will, even the alephs. So maybe you can always ever only get just so high a score... Regardless, the whole matter really is like logic---but it's like perception, too.

EDIT: in a slogan, this is particular knowledge of generality, and general knowledge of particularity. Also, iirc Godel numbering does allow inferences to be conducted in a sort of arithmetical manner, to some extent. So the idea that various set operations correspond to logical forms is not amiss either (for example, think of logical conjunction as the prototype/archetype of the positive hyperoperator sequence; note that you can't apply "x ^n x = x ^n+1 2" to the base, because even if you convert succession into a binary operation, 0 successor 0 does not equal 0 + 2).

Answer 5

Mathematics and logic are two disciplines which depend on each other. Logic is applied to mathematics and math can be applied to logic. The concepts are self-explanatories:

• On one hand, the term mathematics essentially means the study of objects (from the greek "that what is to be studied").

• On the other, logic is the study of the rules that govern our thinking. Kant conceives logic as the science of understanding (COPR B76). Notice that it is the counterpart of the former, in the sense that logic is some type of study of the subject.

As shown, mathematics refers to the study of the rules regarding the objects of nature, and logic focus the rules of thinking of the subject (me, us). In simple terms, the rules of math are applied in order to approach the objects of nature. And the way anything is approached is by means of the rules of logic. Math deals with objects as perceived by the subject, logic deals with how does the subject think of objects.

Since the subject usually defines the object (a basic consequence of empiricism), there's no clear separation between the object and the subject. A rainbow does not exist in nature per se. In order to exist, a rainbow needs of eyes that interpret photons as colors, it needs of a body taking some geographical position, etc. In other words, a rainbow is made in part by the subject (Kant's "things as they appear"), and in part of the object itself (Kant's "thing-in-itself"). The case of rainbows is chosen due to its simplicity of understanding; in fact, any object of nature has equivalent properties, just at different orders of magnitude.

So, if we're going to study rainbows, separating the logic from the math is quite difficult. We can use math to make operations with rainbows, apparently excluding the subject (although the subject defines the object). We can also study the rules of logic, apparently excluding the object (although in such case the subject becomes the object of study). But any non-trivial analysis of the problem will end up in a discussion of the intimate liaison between the object and the subject, logic being applied to math, and math being applied to logic. This is just a consequence of the set of tautologies our truth is based upon, as Kant suggests.