Synopsis

Synopsis

What is the nature of Reality and how can we be sure about what we know? Do mathematical concepts such as symmetry groups and infinity point beyond themselves to a higher reality? How do we account for the fact that mathematics is so effective in describing the natural world?

Questions regarding reality and its relation to mathematics can be traced back to the philosophy of Ancient Greece, notably that of Plato, who posited that his concept of ideal Forms was the embodiment of absolute reality. He was strongly influenced by his predecessors, Thales and Pythagoras (and his followers) who believed that everything was related to mathematics and everything could be predicted in rhythmic patterns or cycles. The concept of numbers and their relationships have long being regarded as examples of Plato’s Forms. The Pythagoreans essentially believed that numbers were both living entities and universal principles permeating from the heavens. In other words, they had a tangible physical existence.

It not surprising that 17^th Century philosopher and mathematician, Descartes proclaimed explicitly that essence of science was mathematics. He regarded the objective world to be “space solidified or geometry incarnate” whose properties are deducible from the first principles of geometry. Galileo, who is largely regarded as the father of modern science, like Descartes, was certain that nature is mathematically designed, for he wrote:

“Philosophy [nature] is written in that great book which ever lies before our eyes – I mean the universe – but we cannot understand it if we do not first learn the language and grasp the symbols in which they are written. The book is written in the mathematical language, and the symbols are triangles circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wonders in vain through a dark labyrinth. ”  

To him, nature is simple and orderly and its behavior is governed by perfect and immutable mathematical laws.

Our quest to uncover and understand nature’s laws, since, has been guided by this believe - that they are mathematical and that they lend themselves to the rules that undergirds mathematical structures. Indeed, the deductive aspect of mathematics has provided a basis for revealing aspects of nature that are not immediately obvious. For instance, Maxwell’s mathematical manipulations of the equations that govern the phenomenon of electricity and magnetism led to the discovery of radio waves. Or for that matter, appreciation that symmetry underlies the way nature behaves at the most fundamental level has led to discoveries of many elementary particles. The most recent being the Higgs particle. One could reasonably argue that twentieth-century physical theories of spacetime, and the interactions that govern elementary particles merits calling the century the “Century of Symmetry”.

The point here being that the study of symmetry necessitate mathematical structures that underlie group theory. It is difficult overstate the significance of group theory in the development of modern physics. Indeed, group theory is the mathematical language of symmetry and it so important that it seems to play a fundamental role in the very structure of nature. It serves as an organizing principle underlying the dynamics of elementary particles to the extent that it appears to dictate the basic laws of physics.

The ubiquitous role that mathematics play in the natural laws has led some physicists to question its use as merely that of a language. Nobel Laurette Eugene Wigner, in his now famous quote: “The unreasonable effectiveness of mathematics” mused that

“The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

This has led some to speculate that our reality isn’t just described by mathematics but it is mathematics. For instance, physicist Max Tegmark, in his Mathematical Universe Hypothesis has put forth that: “Our external reality is a mathematical structure.”   

Learning Outcomes

Learning Outcomes

This is an Inquiry-Tier module offered under USP’s structured collection of multi-disciplinary modules.  Aimed at exposing students to various aspects of scientific inquiry, this module examines issues related to reality and the role that mathematics play in our conception of the former. More specifically the module will address issues such as: 

• What is the nature mathematical knowledge? Or what is the nature of mathematical truths?
• Is mathematical knowledge the same as scientific knowledge?
• How do mathematicians uncover mathematical truths?
• Are mathematicians like scientists discovering pre-existing phenomenon? Or are they like artists starting off with a blank canvass which they fill as they choose? 
• How and why is mathematics so effective in describing the physical world?
• Is mathematics infallible?
• Is mathematics discovered or invented?

In this module, students will be brought through a general framework for thinking about these issues lensed from philosophy, mathematics and physics. Specifically, students will

• appreciate and understand the philosophical underpinnings of the notions surrounding reality and the related epistemological questions;
• examine what constitutes the Axiomatic Method and the use of mathematical reasoning in uncovering mathematical truths;
• learn how mathematics is used in formulating physical laws; and
• examine the philosophical positions that mathematicians take in their views on mathematics.  

Syllabus

Syllabus

Principally, the pedagogical components will be divided into four units:

UNIT 1 - Philosophical and Historical Underpinnings (2 weeks)

The discussions here will examine the ontological and epistemological questions that one is faced with in describing nature. Here, discussions will be grounded in the issues related to reality and the traditions that have been adopted in uncovering this reality. Here, key distinctions between rationalism and empiricism will be discussed.  This will include, among other things, Aristotle’s Inductive-Deductive approach where observations play a fundamental role in knowledge generation. This will be contrasted with Plato’s orientation which favors the contemplation of abstract ideas over sense experience in deriving knowledge.

UNIT 2 – Mathematical Knowledge (3-4 weeks)

This unit will examine some key concepts that underlie mathematics – for instance the move towards abstraction and the use of the Axiomatic Method. These issues will be lensed through some rudimentary number theory and Euclidean geometry. Some basic formal logic will also be introduced to illustrate how mathematicians generate mathematical knowledge through proofs and contradictions. Students will also be exposed to the mathematical structures that underlie algebraic systems and groups as means of illustrating abstraction and dealing with patterns.   Finally, the issue of Godel’s incompleteness theorems will also be discussed to illustrate the point that the commonly held believe that mathematics is infallible is a misnomer.  

Unit 3 – The role of mathematics in the natural laws (3- 4 weeks)           

This unit will examine how the science underwent mathematization and the role mathematics play in the natural laws. Emphasis will placed on how symmetries have provided a guiding principle in unravelling physical processes that define our world. The relationship between symmetries and conservation laws will be elucidated. Some representative questions that will be addressed: How do we represent a physical theory? What constitutes a good theory? How regularities in the laws reflect underlying symmetries in the laws. How symmetries relate to our physical reality.

Unit 4 – Meta issues related to Mathematics (4 weeks)

This concluding unit will examine various philosophical positions taken by mathematicians with regards to mathematics. These viewpoints will be compared and contrasted. Finally, the question:  “Is mathematics invented or discovered?” will serve to get students to reflect on the issue of reality as viewed from mathematics. 

Readings/References

Readings/References

UNIT 1 - Philosophical and Historical Underpinnings (2 weeks)

Michella Massimi, Are Scientific Theories True, in Philosophy for Everyone, editors:   Matthew Chrisman and Duncan Pritchard, Routledge (2014) Pgs – 89 – 106.

Robert G. Olson, A Short Introduction to Philosophy, Dover Publications (1967) - (Chapter 2 – The Nature and Existence of the External World) Pgs. 19 – 35.

Ibid, Chapter 3 – Some representative theories of knowledge; Pgs. 37 – 56.

Ibid, Chapter 4 – The speculative philosophies of Plato and Aristotle; Pgs. 57 – 72.   

UNIT 2 –Mathematical Knowledge (3-4 weeks)

Ian Stewart, Concepts of Modern Mathematics, Dover Publications (1995),  Chapters 2 – 9.

Mario Livio, Is God a Mathematician, Simon and Schuster (2009), Chapter 5.

James Bradley and Russell Howell, Mathematics, HarperCollins Publishers (2011), Chapters 3, 6..

Tony Crilly, The Big Questions – Mathematics, Quercus Publishing Plc (2011), Pgs. 8 – 74.

Unit 3 – The role of mathematics in the natural laws  (3- 4 weeks)

Leon. M. Lederman and Christopher T. Hill, Symmetry and the Beautiful Universe, Prometheus Books (2004), Chapters 4 and 5.

Stephen Haywood, Symmetries and Conservation Laws in Particle Physics, Imperial College Press (2011), Chapters 1.

Max Tegmark, Our Mathematical Universe, Borzoi Books (2014), Chapters 9 and 10. 

Unit 4 – Other Interpretations of Quantum Mechanics (2-3 weeks)

Tony Crilly, The Big Questions – Mathematics, Quercus Publishing Plc (2011), Pgs. 172 – 190.

James Bradley and Russell Howell, Mathematics, HarperCollins Publishers (2011), Chapters 9, 10.

Assessment

Assessment

The assessment scheme for this module is as follows:

This module will be conducted as a 2-hour discussion-led seminar twice a week.