How Math Is Different

The Birth of Modern Mathematics

The Big Questions

1. What Is Mathematics?
2. How does math relate to other sciences?
3. What about rigor?
4. What are the building blocks?

Euclid (300 BC)

Building blocks of Ancient Greeks math:

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• Numbers: 1, 2, 3…
• Magnitudes: 20cm, 3.5kg
• Shapes: points, lines, solids…

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For all these: Axioms, Definitions, Proofs

From 300 BC to 1500 CE

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• Numbers and magnitudes => Numbers
• Axioms and proofs: Geometry only
• Zero (in Europe by 1200)
• Negative numbers (in Europe by 1500)

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16th Century

Complex numbers

i² = -1

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17th Century

René Descartes (1596-1650)

• Use x,y,z for unknowns
• Concept of function: y = f(x)
• Analytic geometry

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Could Greeks have this?

Could Greeks solve x²+x=20?

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18th Century

Complex numbers used to solve problems.

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Leonhard Euler (1707-1783)

19th Century, 1800-1850

Geometric interpretation of complex numbers

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19th Century, 1800-1850

Non-Euclidean Geometry

Nikolai Lobachevsky, János Bolyai,

Bernhard Riemann

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• Axiomatic method
• But axioms are not necessarily true!
• Axioms are definitions in disguise (Poincare)

The Big Questions, circa 1880

What Is Mathematics?

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Study of numbers and shapes.

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The Big Questions, circa 1880

How does math relate to other sciences?

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Helps them by exploring

numbers and shapes

The Big Questions, circa 1880

What about rigor?

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Difficult to obtain.

Maybe kills intuition.

The Big Questions, circa 1880

What are the building blocks?

Shapes.

Numbers (real and strange).

Functions.

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Logic in 1860

Charles L. Dodgson (Lewis Carroll) 1832-1898

Logic in 1910

Bertrand Russell, Alfred North Whitehead

Axiomatization of Geometry

David Hilbert (1862-1943)

Axioms for Geometry (1899)

"One must be able to say at all times — instead of points, straight lines, and planes — tables, chairs, and beer mugs"

Set Theory

• Georg Cantor (1845-1918)
• Developed set theory in 1870s

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“A set is the comprehension of given things as a single whole”.

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Infinite Sets

R

Actual Infinity

Potential Infinity

Rich Structure

Real numbers:

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• can add +
• can multiply *
• has order <
• closeness
• denseness

Rich structure

Complex numbers:

• can +, can *, no order

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Vectors:

• can +, no order, no *

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Axiomatic Method

Isolate some relationships

Define their axioms

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Define a structure by them

Axiomatic Method, 1880-1930

• Slow exploration
• New structures added when they solve real problems
• Vector space: 1888… 1906… 1920, finally!

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• Probability axioms (Kolmogorov): 1931.
• Instant success

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Two Algebra textbooks

Weber, 1898, Algebra:

1 word “Axiom” in the book

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van der Waerden, 1930, Moderne Algebra:

251 words “Axiom” in the book

The Big Questions, circa 1930

What Is Mathematics?

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Study of abstract patterns

The Big Questions, circa 1930

How does math relate to other sciences?

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Freed from dependance Relaxed field

Often helps them anyway

The Big Questions, circa 1930

What about rigor?

Basic structure has rigor

Advanced stuff difficult

Intution still exists

The Big Questions, circa 1930

What are the building blocks?

Abstract structures: sets of elements

Axioms determine relationships

Elements don’t matter

Structure matters

Axiomatic Method

• Inspired by Ancient Greek axioms
• and non-Euclidean geometry
• and rigorous logical proofs
• and sets as single separate objects

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Structure = set of elements with relationships defined by axioms

After 1930

Nicolas Bourbaki: 1930s- 1960s

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Category Theory: 1930-1950

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Math and Sciences

“...it is clear that the applications to analytical physics must be extensive in a high degree…”

William Rowan Hamilton, 1843

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On The Unreasonable Effectiveness Of Mathematics in the Natural Sciences

Eugene Wigner, 1960