RI Math Lecture 3 2023

MathematicsLecture 3: Discovered or CreatedMath’s effectiveness in explaining Nature

OverviewuDiscovered or Created? Real or Non-real? Objective or Subjective?uMath and the Natural WorlduWhy is Math so Effective? The OptionsuMath and other fields

Discovered or Created? Real or Non-real?uOne of the key questions in any Area of Exploration (Math, Science, Social Science, Ethics, Aesthetics)uGeneral definition of Realism: “a,b, andcand so on exist, and the fact that they exist and have properties such asF-ness,G-ness, andH-nessis (apart from mundane empirical dependencies of the sort sometimes encountered in everyday life) independent of anyone’s beliefs, linguistic practices, conceptual schemes, and so on.” (SAP, “Realism”)uThere are thus 2 general aspects to realism:u1) Existence –e.g. tables, rocks, the moon etc all exist, as well as facts about the table (it is square), rock (it being made of granite) and the moon (spherical)u2) Independence –e.g. the fact that the moon exists and is spherical is independent of anything anyone happens to say or think about the matter, as with the table being squareuE.g. A realist about the existence of tables is someone who believes that tables exist independent of human minds such that it has its properties regardless of what people think or say, i.e. objectiveuOn the other hand, a non-realist denies one or both dimensions of Realism.uNon-realism can take many forms and anti-realism is a subset of non-realism

Realism and Non-realism about MathuA realist about Math is someone who believes that mathematical objects exist independently of human minds and has its properties regardless of what people think or sayuA realist about Math thus thinks that Math is to be discovered, i.e. math is objectiveu2 main schools: Platonism (including Logicism) and Empiricism uAn anti-realist is someone who concedes the existence aspect of realism but denies the independence aspect, i.e. math is created and subjectiveuSo an intuitionist believes that mathematical objects do exist but only because they are constructed by the human mind and that is why it has the properties it hasuA nominalist (e.g. formalism) on the other hand is someone who denies the existence dimension; mathematical objects are mere vocal utterances

Discovered/Objective vs Created/SubjectiveuSo the question of Discovered/Real vs Constructed/Non-real is also a question of Objective vs SubjectiveuOf course, there is a third possible option: Kantian view on Math, i.e. intuitionismuHere, Math is objective not because of the independence dimension but because it is universally created due to a shared mental apparatus, i.e. Filters of ConsciousnessuThere is thus no biasednesssimply because it is ‘universally’ biaseduNote: this notion of “Universality” for intuitionism/Kantian mathematics merely applies to humanminds, NOT all mindsuIn other words, it is not strictly universal in the traditional sense like what Platonism would holduSo it is entirely possible that ali