There is an almost ubiquitous misconception about what mathematics really is, and it’s a misconception that genuinely beckons a correction. I would take a guess that if the average person was asked “What is mathematics?”, they would respond with something along the lines of “well, it’s a bunch of rules that help you find certain numbers”. While this may have been a correct answer long ago, it is far from correct today.
Fortunately, what it actually is can be summarized very succinctly. Mathematics is simply the process of making assumptions and proving what follows. Hence, we all do math on a daily basis—either when talking to one another, or when thinking to oneself: “given what I know, I think that…”. This is math.
This is also why math courses through calculus are terrible—as they are absurdly misleading. Current curriculum is libelous to the discipline of mathematics and its participants, and action needs to be taken to address this.
At this point, high school language arts and composition courses may teach more math than actual math courses. Fundamental to mathematics is logic and its application in the context of sets. Logic was essentially nonexistent when I was in high school. Yet math and language arts classes implicitly assume that students have a solid understanding of it when they are asked to make arguments. Granted, as we are logical entities cognitively, we are trivially masters of logic. But in terms of conveying it communicably, improved training is necessary. I feel the overlooking of this necessity is a grave miscalculation that has hindered scientific thinking (an ability from which every citizen of the world can drastically benefit) for far longer than it should have. This needs to change.
Recall the definition of an -ary logic system as a homomorphism with a theory, a structure, and a structure of cardinality such that and have the same signature. Fuzzy logic is hereby defined as the study of -ary logic systems where is the cardinality of the continuum. In this sense propositions can be thought of as having valence values in some interval, like So classical binary boolean logic is a type of fuzzy logic, where
Example 1. Recall in our construction of a utilitarian set we had a set together with a utility function Now suppose is a set of terms which is also a utilitarian set. Suppose we define a structure on by
This structure has signature Then is also a set of terms, with certain terms in identified, and if is a theory/subset in then is a -ary logic system where has the min and max operations.
In the case of Example 1, if we think of the set of terms as a set of behaviors, which could be construed as terms of persons (acting as words), then the structure can be interpreted as equivalence classes of compound behaviors that yield the same utility where logical valence of compound behaviors is simply based on their utility.
Since the valence set in a fuzzy logic system is an interval, let us look at some common structures on to discuss some intuitive structures on terms. We already mentioned min and max functions and a corresponding structure on class utilitarian sets. is closed under binary multiplication. The corresponding binary connective would thus send two propositions to a proposition whose valence is a product of the original two. Treating binary boolean logic as a fuzzy logic, propositional conjunction satisfies this condition.
Example 2. Let be a probability space. Let be closed under addition where any sum exceeding is defined as and define a unary operation by Now note that is closed under union and complementation; denote this structure on by If measurable sets are construed as formulas, select a theory of disjoint sets. Then is a fuzzy logic system.