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 alien minds would construct math differently from us due to different mental apparatus

Do mathematical objects exist in the Natural World?uAnswer: No. An actual, i.e. mathematically correct, circle doesn’t exist in the physical world. Coins, wheels, coin prataetc. are only approximationsof actual circles.uNo matter how good a drawing you have, even by a computer, it is not a mathematically correct circle –or any other geometrical object for that matter.uAt some level of magnification, you can see that the ‘circle’ drawn by the computer is just a series of pixels that lie next to each other. uEven something as simple as a line, defined as that which has length but no breadth, is impossible to draw correctly. uIn other words, mathematical objects do not exist in the natural world. Rather, they seem to exist in another realm (or have no existence whatsoever)uPlatonists and Logicists: Platonic heavens (Discovery) uIntuitionists: in our minds (Construction).uFormalists: such mathematical objects have no existence per se; they are merely the rules of a manmade game.

Math and the Natural WorlduYet Math seems to be really effective at explaining the workings of the natural world and it has real practical uses! Here are some examples:ui) Ellipses –3rdC BC development by the Greeks. Had no practical purpose, was pursued purely for intellectual interest at that time BUT was put to use by Johannes Kepler to explain planetary motion in the 17thC AD! (Unreasonable Effectiveness, 6)uii) Calculus is also used to explain planetary motionuiii) Euclidean geometry –used for construction and navigationuiv) Riemannian geometry –developed by Riemann as a purely intellectual exercise but 30 years later, Einstein concluded that space conforms to Riemannian rather than Euclidean geometryuv) The advancement of physics beyond classical mechanics to cover electricity, magnetism, sound and light waves were all made possible by corresponding developments in the theory of ordinary and partial differential equations.uvi) And of course, so much of physics is captured in mathematical equations! E.g. f=ma, p=iv, f=ke, e=mc2uAfter all, if physics is the study of the laws of nature (LON) and LON are simply conditional statements of how the world works, to predict what is going to happen in the future, it is extremely helpful if these LON are formulated in mathematical language –you get a level of precision that would not have been possible otherwise.

A side note…uNote 1: actually, these mathematical models merely approximatephysical reality.uE.g.: Planets don’t move in perfect ellipses, the land is not perfectly flat like in Euclidean planar geometryuNonetheless, Newton’s law of gravity was proved accurate to less than a ten thousandth of a per cent (Unreasonable effectiveness, 6) uNote 2: the relationship between Physics and Math is intricate and mutual. It is symbiotic. Not only does math contribute to physics, physics does too. uThere are instances of developments in physics motivating new areas and results in math such assymmetries in physics spurring the growth of group theory, brownianmotion leading to functional integrals or the use of non-abelian gauge theory combined with supersymmetry in particle physics setting the stage for important work in modern math. (Math and the Real world (364)

Math and the Natural WorlduThe question is “WHY?” uWhy is Math, an abstract, mental and intellectual exercise, with its own internalised and idealised logic that depends not on the natural world, so applicable to the real, i.e. physical and natural world?uOr why is Math so “unreasonably effective” in its explanations of and application to the natural world?u“How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” –EinsteinuHow does it happen that a subject like mathematics, seemingly constructed and policed entirely by the ‘inner world’ of human minds, ends up being such a successful tool in describing and indeed harnessing the external physical world? Is it that the physical world has some intrinsic ‘mathematical order’, which then instilled in human brains the basic concepts of mathematics and logic through the evolutionary process? In other words, did the human mind learn about mathematics from the external world rather than the other way around[i.e. instead of trying to learn about the external world frommathematics]?R. Rajaraman, emphasis hisMathematics and the real world, 361

Why is math so effective?uSeveral options are open to us:u1) The natural world itself is mathematical and we discover math through it, i.e. empiricismu2) The natural world is a ‘copy’ of the mental world where math resides as one of the Forms, i.e. platonismand logicismu3) The natural world is neither mathematical nor a copy of some mental world but simply appears to us to be mathematical as we imposeour mathematical understanding onto the natural world, i.e. intuitionismu4) Th