Foundations of Mathematics

Mathematical and Formal Logic

What are some philosophies of mathematical logic?

Over the centuries, there have been many mathematicians who have tried to figure out the complexity of mathematics. Fast-forward to modern times, and one finds that mathematical knowledge and logic have dramatically changed. In particular, in the late 20th and early 21st centuries, the development of predicate calculus and the digital computer has had a great impact on these studies. And out of these ideas—not to mention centuries of mathematics and logic groundwork—are some of the more interesting philosophical doctrines of mathematical thought:

  • Formalism is the idea that mathematics is truly formal; therefore, it is only concerned with the algorithmic manipulation of symbols. In formalism, predicate calculus does not denote predicates—or anything else—meaning mathematical objects do not exist at all. This definitely fits into today’s world of computers, especially in the field of artificial intelligence. But this philosophy does not take into account human mathematical understanding, not to mention mathematical applications in physics and engineering.
  • Set-theoretical Platonism sounds as if mathematicians are regressing back to Plato’s time. In reality, this philosophy is based on a variant of the Platonic doctrine of recollection, in which we are born possessing all knowledge and our realization of that knowledge is contingent on our discovery of it. In the set-theoretical Platonism, infinite sets exist in a non-material, purely mathematical realm. By extending our intuitive understanding of this realm, we can cope with the problems such as those encountered by the Gödel incompleteness theorem. But this philosophy, like the others, has a seemingly infinite number of gaps, especially the question of how can a theory of infinite sets be applied to a finite world.
  • Constructivism was a “fringe” movement at the turn of the 21st century. Constructionists believe that mathematical knowledge is obtained by a series of purely mental constructions, with all mathematical objects existing only in the mind of the mathematician. But constructivism does not take into account the external world, and when taken to extremes, it can mean that there is no possibility of communication from one mind to another. This philosophy also runs the chance of rejecting the basic laws of logic. For example, if you have a mathematical problem with a yes or no nature, with the answer unknown, neither “yes” nor “no” is in the mind of the mathematician. This means that a disjunction is not a legitimate mathematical assumption—and thus, ideas such Aristotle’s law of the excluded middle (“either or”) are cast aside. And not many modern mathematicians want to throw out centuries of logic.
  • Structuralism holds that mathematical theories describe structures—and that mathematical objects are defined by their place in such structures, but have no intrinsic properties. For example, if one knows that the number 1 is the first whole number after 0, then that is all that needs to be known. Even though it sounds simple, structuralism only relates to “… what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have…” Thus, it has some interesting interpretations in philosophy.
  • Fictionalism sounds like something you read that’s the opposite of non-fiction, but it’s actually a type of mathematical philosophy proposed around 1980. But in a way, the mathematician who brought out the proposal of fictionalism, Hartry Field, was talking about fiction after all: He believed that mathematics was dispensable, and should be considered “…as a body of falsehoods not talking about anything real.” This type of logic has not been readily accepted, as there are some logic and statement problems inherent in his philosophy.
  • Social realism, or social constructivism, proposes that mathematics is mainly a construct of culture that is subject to change or even correction. It’s often said that this idea is the opposite of how mathematicians treat their field—that mathematics is more pure and objective. Social realism theorists believe this isn’t true, that mathematics is grounded by much uncertainty, and grows, like things in nature, through a sort of mathematical evolution.


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