Mathematics: Certainty, Proof, and the Foundations

The Nature and Purpose of Mathematics

• The subject of Mathematics is described with paradoxes and rhetorical quotes that frame its nature, usefulness, and beauty:
  • Mathematics is neither physical nor mental, it is social. (Reuben Hersh)
  • The useful combinations in mathematics are precisely the most beautiful. (Henri Poincaré)
  • Mathematics is the abstract key which turns the lock of the physical universe. (John Polkinghorne)
  • Everything that can be counted does not count. Everything that counts cannot be counted. (Albert Einstein)
  • The mark of a civilized person can be seen in the ability to confront a column of numbers with emotion. (Bertrand Russell)
  • The advancement and perfection of mathematics is linked to the prosperity of the state. (Napoleon)
  • In pure mathematics we contemplate absolute truths that exist in a timeless realm (divine mind metaphor).
  • Some see mathematics as impressions rather than fixed answers, highlighting human subjectivity.
  • A humorous note: Mathematics began when a brace of pheasants and two days coincidentally yield the number two. (Bertrand Russell)
• Introduction to mathematics as certainty and utility:
  • Mathematics is often seen as an island of certainty in a sea of doubt (2+2=4 is trusted;
    314 for a circle's circumference/diameter ratio is constant). $2+2=4$, rac{C}{d}= ext{(approximately }3.14)
  • Galileo: “the book of nature is written in the language of mathematics.”
  • Mathematical literacy is essential for many scientific careers; mathematics provides both certainty and practical value.
  • Bertrand Russell recounted a personal, almost romantic encounter with geometry at age eleven, illustrating a child-like delight in mathematical certainty. This joy may perplex or threaten others who view mathematics as lacking human warmth.
  • The certainty of mathematics can be reassuring to some and threatening to others because mistakes are clear and unambiguous in maths (you are wrong if you are wrong).
• Language, notation, and notational diversity:
  • The Piraha tribe example shows that some cultures lack numerals beyond low counts (one, two, many), yet mathematics as a discipline argues for universality of deeper concepts (e.g.,  as a constant in circle ratio).
  • Notation varies across cultures and bases (base-10 is common due to finger counting; base-12 is argued by some mathematicians to be more convenient because of divisibility). The value of or circle circumference/diameter remains the same across cultures.
  • The question of whether mathematics is universal or culturally contingent is raised (LQ: Cultural perspectives). The symbol language and base systems illustrate cultural influence on notation, while the subject's underlying relationships appear universal.
• The mathematical paradigm and the nature of proof:
  • Mathematics is increasingly described as the science of rigorous proof.
  • Early mathematics included cookbook methods for solving problems; the Greeks (Euclid) formalized mathematics as deductive proof.
  • Euclid’s geometry is the archetype of a formal system; modern high-school geometry inherits this Euclidean framework.
  • The concept of a formal system hinges on axioms, deduction, and theorems, with axioms serving as starting points without proof.
• Activity prompts and reflections:
  • LQ: How does mathematics resemble/differ from natural languages? Is math a universal language?
  • LQ: Is the book of nature written in the language of mathematics?
  • Activity 12.1 prompts reflection on how beliefs about mathematics' value are shaped by one’s competence and engagement with the subject.

The Mathematical Paradigm

• Axioms and the three key components of a formal system:
  • Axioms: starting points accepted without proof; not everything can be proven.
  • Deductive reasoning: deriving conclusions from premises in a logically necessary way.
  • Theorems: statements proven from axioms or previously established theorems.
• Four traditional requirements for a good axiom system:
  • Consistent: cannot derive both p and ¬p from the same axioms.
  • Independent: no axiom can be derived from the others; minimizes redundancy.
  • Simple: axioms should be clear and straightforward.
  • Fruitful: the system should enable proving many theorems with a small axiom set.
• Euclid’s five axioms (a starting point for geometric reasoning):
  • 1. A straight line can be drawn joining any two points.
  • 2. A finite straight line can be extended indefinitely in a straight line.
  • 3. A circle can be drawn with a given center and through a given point.
  • 4. All right angles are equal to one another.
  • 5. Through a given point, there is exactly one straight line parallel to a given line.
• Concept of a point and a line as primitive notions; geometry builds from these with axioms to prove theorems.

Deductive Reasoning and Theorems

• Deductive reasoning as a chain from general premises to particular conclusions.
• Example theorems in Euclid’s system:
  • 1. If two lines are perpendicular to the same line, they are parallel.
  • 2. Two distinct straight lines do not enclose an area.
  • 3. The sum of the angles of a triangle is 180^\u00b0.
  • 4. The angles on a straight line sum to 180^0.
• A classic proof structure (Figure 12.4): given a + c = 180°, and a + b = 180°, substitute to conclude b = c (QED).
• Generality of proofs: a proof can be applicable to any scale, not just a specific instance.
• Equalities, substitution, and logical progression are core components of proving theorems.

Proofs and Conjectures

• Distinction between proofs and conjectures:
  • Proof: a theorem is shown to follow logically from axioms.
  • Conjecture: a hypothesis that seems to work but is not yet proven.
• The sum of the first n odd numbers equals n^2:
  • Empirical check for small n appears to hold; this is an inductive demonstration, not a proof by itself.
  • Induction: reasoning from specific cases to general truth cannot guarantee certainty in all cases (black swan analogy).
• The general caution: induction cannot provide certainty about all future cases.
• Goldbach’s Conjecture (even numbers as sums of two primes):
  • Verified computationally for very large ranges (up to 10^14 in examples).
  • Despite extensive verification, a full proof remains elusive; the possibility of a counterexample at an enormous scale cannot be ruled out.
  • Wittgenstein’s warning about equating large numerical testing with certainty: tested cases are infinitesimal relative to infinity.
• The idea of a proof being a demonstration that follows from axioms, contrasted with empirical validation and computational verification.

Beauty, Elegance, and Intuition

• Insight and elegance play a role in mathematical reasoning:
  • Many elegant proofs are valued for being clear, economical, and beautiful; Erdos’s idea of the BOOK of beautiful proofs illustrates a cultural ideal in mathematics.
• The value of intuition in mathematical creativity:
  • Intuition can guide problem-solving and discovery, but educated intuition is distinguished from natural intuition; Mathematica-like fireworks must be supported by proof.
• Activity 12.4: proofs that reveal elegant structures, such as the sum of integers or KO tournament counts, illustrate how reframing a problem can yield simple, powerful solutions.
• Activity 12.5–12.6: mind the role of intuition in problem-solving and the limits of calculators/computers in understanding.
• The educational takeaway: educated intuition matters, but proof remains the gold standard for mathematical certainty.

The Social Dimension and Technology

• Modern proofs often span hundreds of pages and require collaboration; trust in peer verification is essential.
• The role of computers in mathematics:
  • Computers help discover patterns in data and aid in proofs, but raise concerns:
  • Buggy programs can undermine proofs.
  • Outsourcing understanding to machines may erode our own grasp of the concepts.
• The social nature of proof: The test of a proof is whether qualified judges are convinced; this social criterion governs mathematical validation.

The Nature of Mathematical Certainty: Analytic vs Synthetic, and a Priori vs a Posteriori

• Fourfold matrix of propositions:
  • Analytic vs Synthetic (true by definition vs not true by definition).
  • A Priori vs A Posteriori (knowable independently of experience vs only through experience).
• Box 1: Analytic and A Priori — definitions; e.g., Bolivian bachelors are unmarried (trivial truth learned independently of experience).
• Box 2: Analytic and A Posteriori — empty because analytic truths can be known independently of experience; the combination is self-contradictory.
• Box 3: Synthetic and A Posteriori — empirical knowledge about the world (e.g., elephants in Africa).
• Box 4: Synthetic and A Priori — non-trivial truths knowable independently of experience; the big question for mathematics.
• The central question: Where does mathematics fit?
• The three options about the status of mathematics:
  • Option 1: Mathematics as empirical (Mill): mathematical truths are empirical generalisations based on many experiences; the certainty of arithmetic is due to repeated confirming instances.
  • Option 2: Mathematics as analytic (true by definition): arithmetic truths are unpacking of definitional content; e.g., 2+2 = 4 is just restating a definition.
  • Option 3: Mathematics as synthetic a priori: mathematics provides non-trivial knowledge about reality that can be known a priori; historically linked to Euclid and classical geometry.

Three Philosophical Views: Empiricism, Formalism, Platonism

• Empiricism (Option 1): mathematical truths are generalizations derived from experiences; Mill’s view uses induction from many instances to claim certainty, but has limits.
• Formalism (Option 2): mathematics is a game of symbols; truths are true by definition within a chosen formal system; the world’s truths depend on axioms chosen, not on an external reality.
• Platonism (Option 3): mathematics reveals a priori truths about an abstract realm; mathematical objects exist independently of human minds; timeless and certain.
• The text notes that in practice, philosophers and mathematicians espouse mixtures of these views, and debates persist about certainty and truth in mathematics.

Discovered or Invented? The Platonism vs Formalism Debate

• Distinguishing the two positions:
  • Platonism: mathematical entities exist independently of us and are discovered; mathematically real even if not physical.
  • Formalism: mathematics is a human-made creation; mathematical truths are true within the rules of a chosen formal system and do not exist outside that system.
• Activity 12.11 prompts reflection on discovery vs invention and asks whether intelligent aliens would develop the same or different mathematics.
• The circle example illustrates idealisations: exact circles do not exist in the physical world; mathematical circle is defined as the set of all points equidistant from a given center, an idealisation that cannot be drawn perfectly on a paper; this raises questions about existence, drawing, and the nature of mathematical objects.
• The dilemma: mathematical objects seem to lie between mental constructs and external reality; Platonists argue for a real, timeless existence; formalists deny independent existence and see math as a human convention.

Non-Euclidean Geometry and the Problem of Consistency

• Euclidean geometry was considered a universal model of knowledge, but one axiom (the parallel postulate) invited doubt about self-evidence and consistency.
• The parallel axiom’s non-obviousness led to attempts to prove it from the other axioms; these efforts failed for centuries.
• 19th century shift: Georg Friedrich Bernhard Riemann proposed alternative systems by altering some axioms:
  • Axiom changes in Riemannian geometry include: two points may determine more than one line; all lines are finite yet endless (curves like circles); there are no parallels.
• The theorems in Riemannian geometry include:
  • Perpendiculars to a straight line meet at a point; two lines enclose an area; the sum of angles in a triangle exceeds 180 degrees.
• The geometric intuition: on a sphere, straight lines are great circles; this helps explain how non-Euclidean geometry can be consistent and meaningful in a curved space.
• The broader lesson: the consistency of a formal system is not guaranteed by intuition or historical success; alternative geometries can be internally coherent and useful in describing reality (e.g., space is curved per Einstein).
• The problem of consistency arises because a system could be free of contradiction while still not reflecting physical reality; thus multiple geometries can exist as valid frameworks for modeling space.
• The barber paradox serves as a cautionary tale about self-reference and the perils of seemingly clear instructions that lead to contradictions; it is used to illustrate limits of intuitive reasoning.

Gödel, Incompleteness, and the Limits of Certainty

• Gödel’s Incompleteness Theorem (1931): no formal mathematical system can prove its own consistency; there will always be true statements that cannot be proved within the system.
• What Gödel implies: absolute certainty in mathematics is unattainable in an absolute sense; mathematics is complete only within formal systems that may be incomplete themselves.
• Practical impact: mathematicians accept the limitations and continue using rigorous proofs within well-defined systems, but acknowledge there may be true statements beyond any given system’s reach.
• The implications for certainty in mathematics: even Euclidean geometry’s certainty is challenged by relativity and the acceptance of non-Euclidean geometries; reality may guide which mathematical framework best describes the world.
• Einstein’s reflections: mathematical systems are invented; which system describes reality is an empirical question; there is an interplay between invention and discovery.

Applied Mathematics and the Unreasonable Effectiveness of Mathematics

• The shift toward applied mathematics: mathematics is used to model real-world problems; the transition from purely abstract to applicable mathematics is highlighted.
• The Fibonacci sequence and the golden ratio:
  • Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
  • Ratios of successive Fibonacci numbers converge to the golden ratio rac{a{n+1}}{an} o rac{1+\u221a5}{2} .
  • The golden ratio (approximately $$rac{1 + 1?}{2} \