A Brief Natural History of Mathematics

1
A Brief Natural History of Mathematics

• James Wysong
• Hillsborough Community College
• Tampa, Florida

2
CHANGING MINDS

• Who are we and why?
• What change(s) transformed us?
• How did the capacity for symbolic thought arise?

3
CHANGING MINDS The discovery and role of FOXP2
gene
4
CHANGING MINDS A Clue FOXP2 Mutations
5
CHANGING MINDS
BROCA and the BRAIN
6
HOMINOID SEE HOMINOID DO

• THE ROLE OF MIRROR NEURONS IN DEVELOPING A
  THEORY OF THE MIND

MIRROR NEURONS and imitation learning are the
driving force behind "the great leap forward" in
human evolution. V.S. Ramachandran
7
SELF AWARENESS AND EMPATHY
8
FROM FEW - MANY

• DNA analysis indicates a bottleneck in human
  population occurred 60,000 years ago
• Estimated world population fewer than 10,000
  individuals
• Homo sapiens limited to Africa

From a genetic point of perspective, all humans
are therefore Africans, either residing in Africa
or in recent exile. Svante Paabo of the Max
Planck Institute
9
THE TOBA SUPERVOLCANO
10
FROM FEW - MANY

• DNA and anthropological evidence points to the
  San people as our common ancestor

11
A NEW BEGINNING

• Evidence of a modern mind
• Art
• Music
• Counting / tally systems

12
A NEW BEGINNING

• Music as a memory device?
• Complex tonal music activates the same neural
  circuits as mathematics, especially geometry

The Mozart Effect ?
13
A NEW BEGINNING

• If all art aspires to the condition of music, all
  the sciences aspire to the condition of
  mathematics. - George Santanaya
• Some songs are just like tattoos for your
  brain... you hear them and they're affixed to
  you. - Carlos Santana
• Music is the pleasure of the human soul
  experiences from counting without being aware
  that it is counting. - Gottfried Leibniz

14
A NEW BEGINNING

• A 30,000 year old lunar calendar?
• Symbols and counting applied to the study of the
  sky

Days are numbersWatch the starsWe can only see
so farSomeday, you'll know where you are Lyrics
from The Alan Parson Project
15
FROM GENES TO MEMES

• Richard Dawkins coined the term meme, which first
  came into popular use with the publication of his
  book The Selfish Gene in 1976
• Memes are a unit of cultural information
  transferable from one mind to another. Examples
  of memes are tunes, catch-phrases, clothes
  fashions, ways of making pots or of building
  arches.

16
PROXIES FOR REALITY
Without Memes, your Genes cannot tell you what
any of these are.
17
OUT OF EDEN
He saw the great Mystery, he knew the HiddenHe
recovered the knowledge of all the times before
the Flood.He journeyed beyond the distant, he
journeyed beyond exhaustion,And then carved his
story on stone.
From the Epic of Gilgamesh ca. 2700 BCE
18
A NEW WAY TO REMEBER

• Was civilization a consequence of climate
  change?
• Complex economies required accountancy
• Mathematics and writing arise as the tools of the
  accountants and bureaucrats

19
A NEW WAY TO REMEBER
Dr. Denise Schmandt-Besserat describes in her
book How Writing Came About the evolution of
symbolic tokens to writing and number symbols.
It took no fewer than four inventions tokens,
envelopes, markings and tabletsand about 4,000
years to fully reduce three-dimensional tokens
for counting to written signs.
20
A NEW WAY TO REMEBER

• Civilizations require more complex
  counting/recording systems
• These systems then foster new developments in
  society
• With civilization, mathematics and writing
  co-evolve

21
A NEW WAY TO REMEBER

• Meso-American cultures offer a test of this idea
• The Mayan civilization, independent of Eurasian
  cultures, developed sophisticated mathematics and
  a writing system

22
LETTERS, NUMBERS, AND CIVILIZATION

• Clocks and Calendars

23
LETTERS, NUMBERS, AND CIVILIZATION

• Navigation

24
LETTERS, NUMBERS, AND CIVILIZATION

• Astronomy
• Astrology

25
FAITH IN NUMBERS

• The Greeks plant the seed of Science by applying
  mathematics to the study of nature

Aristotle (384-322)
26
FAITH IN NUMBERS
Increasingly complex and powerful mathematics
develops

• Axiomatic method
• Number theory
• Three dimensional geometry

The student asked What shall I get by learning
these things?" Euclid called his slave and said,
"Give him threepence, since he must make gain out
of what he learns."
27
FAITH IN NUMBERS

• The power of mathematics and logic to study the
  world.

Eratosthenes of Alexandria (c.275-192 BCE)
determines the circumference of the Earth with
reasonable accuracy
28
FAITH IN NUMBERS

• The Geocentric model of the Cosmos as refined by
  Claudius Ptolemaeus

Ptolemys system was the accepted paradigm for
sixteen centuries
29
NEW NUMBERS NEW IDEAS

• The introduction of Hindu-Arabic numerals by Al
  -Khwarizmi brought profound changes to mathematics

He wrote his Al-jabr wa'l muqabala (from which
our modern word "algebra" comes). The English
word "algorithm" derives from the Latin form of
al-Khwarizmi's name.
30
NEW NUMBERS NEW IDEAS
The Islamic Golden Age 750 to 1050 A.D.

• The consolidation and preservation of Greco-Roman
  knowledge
• The development of Algebra
• Advances in number theory, trigonometry,
  geometry, and physics

"Verily the men of knowledge are the inheritors
of the prophets", (Hadith)
31
THE WHEEL IN THE SKY

• In 1543 Copernicus publishes De Revolutionibus
• The scientific renaissance is underway

The idea of Heliocentrism was so radical the term
revolution took on a second meaning dramatic
and profound change.
32
THE WHEEL IN THE SKY

• In 1609 Kepler publishes the first of his laws of
  planetary motion
• He was also first to derive logarithms purely
  based on mathematics
• His method of infinitesimals foreshadowed calculus

33
THE WHEEL IN THE SKY
Galileo Galilei Professor and chair of
mathematics at the University of Padua

• Pioneered the use of quantitative experiments
  with results analyzed mathematically
• Establishes Physics

34
THE WHEEL IN THE SKY
Sir Isaac Newton Lucasian professor of
mathematics Cambridge University

• In the Principia Newton uses only 59 words to
  describe motion in the Cosmos
• Invents Calculus as a tool
• Establishes a Clockwork Universe paradigm

Let mortals congratulate themselves that so
great an ornament of human nature has existed.
Inscription on Newtons monument
35
FROZEN FIRE AND THE LIMITS OF CERTAINTY
Albert Einstein

• Relativity spells the end of the Clockwork
  Universe
• Emc2 A marriage of the conservation laws and
  so much more

36
FROZEN FIRE AND THE LIMITS OF CERTAINTY
From Special and General Relativity come
"One thing I have learned in a long life  that
all our science, measured against reality, is
primitive and childlike--and yet it is the most
precious thing we have."    - Albert Einstein 
A new view of gravity A new view of time and
space The origin of the Universe from a
singularity
37
FROZEN FIRE AND THE LIMITS OF CERTAINTY

• The expansion of the Universe predicted by
  General Relativity is observed by Hubble in 1929

Georges Lemaitre used Einsteins work to predict
Hubbles discovery in 1927
38
FROZEN FIRE AND THE LIMITS OF CERTAINTY
The Architects of Modern Physics 1927 Solvay
Conference
39
WHAT NEXT?

• Your home work assignment
• Develop a quantum theory of gravity

For perfect happiness the intellect needs to
reach the very Essence of the First Cause. And
thus it will have its perfection through union
with God... St. Thomas Aquinas
40
Thank you for your kind attention
Jim Wysong