MAP
- all associated WEB PAGES of Robert A. Hendrix, MD, FACS
Robert Hendrix & Mathematics From the View of the Hawk
or
Numerical trysts from my love of mathematics
- Confessions of a Sets Maniac...
featuring the on page movie:
Coyoteman Isomorph
= see what happens to me when the sun goes down...
Last updated September 21, 2004
Mathematics From the View of the Hawk
or
Numerical trysts from my love of mathematics
Confessions of a Sets Maniac...
The Mensch who flies higher sees further?
Certainly, from afar, less cumbersome detail clutters the vision.
However, insufficient detail can render the familiar unrecognizable
And flying too close to the sun can have mythical consequences.
All contents of this webpage including text, images and my musical composition,
Erika's Song (copyright 1988) are copyrighted -- all rights reserved; March 2002
Generalizations of some of my favorite mathematical ideas
Nothing is something
Mathematicians can make a great deal of nothing.
"Vacuum" is an 'empty' concept.
Similarly, "nothing" is an 'empty' concept,
However, a set containing nothing is something.
Natural numbers are defined from sets of sets of the empty set, = { }.
Counting up (even birds can count up to three but then they get lost in "many"...)
1 = {}
2 = { , {}}
3 = { , { }, {{ , {}}}}
... und so weiter
n +1 = n {n}
study this pattern a bit and it becomes apparent that
n+1 = {1} {2} {3}
... {n-3}
{n-2} {n-1} n
where each natural number, 1, 2 3,..., n-3, n-2, n-1, n is actually a set as defined above.
So counting and ordering discrete objects are based on sets of sets
... maybe you have a better idea?
Don't shoot me; I'm just the 'Peano Player'
Giuseppe Peano (1858-1931) was a late 19th century logician who influenced Bertrant Russell
We offer the New World 'Order'!
- Something for Nothing based on sets of sets Admittedly, doing arithmetic with sets of sets is about as inviting as
computer programming in machine language as did
Uncle John von Neumann.
(caricature by Andy Hendrix)
So just use
the symbols, 1, 2, 3,... for which you already have developed plentiful dendritric neuronal processing
Never ask me exactly what a set is because I likely won't answer…
"Set" is not defined
Nonetheless the language of Set Theory is the footings for the Castles of Math
-- nearly EVERYTHING is expressable in terms of set theory.
Nothing scares a mathematician like INCONSISTENCY
but it certainly didn't bother some of the women I've known in my life...
To avoid inconsistency, do not define your most basic concepts.
Define the undefined as 'primitive terms' and ask no more
"There are things man was probably not meant to know exactly"
Heisenberg might have said something like that
and Neils Bohr might have answered, "That's crazy talk, Werner!"
Heisenberg or Neils might even have been here
... But probably not
DEFINING EVERY SINGLE CONCEPT (too rigorously) IS THE FOOL'S ROAD TO PARADOX, KENTUCKY.
Leave a few terms undefined and 'primitive'
To celebrate your ancestral origins
(just as your appendix celebrates your ancestral organs)
If it is too definite what a set is,
it might seem possible to contain God as an element of a set
Nietsche would have called that the null set,
(also known as the number "1")
...Nietsche died of 'brain fever' for some ungodly reason...
Polytheists would argue that the God Set had more than one element
Pantheists would maintain that the God Set was THE SET OF ALL SETS
Halleluia! Halleluia! Hal - le -lu - ia !
Mystics would assert that We are ALL elements of the God Set
Theists would employ Venn Diagrams to analyze whatever they analyze
Satanists would wonder about the attributes of the set 'complementary' to the God Set
Agnostics would regard the God Set as unknowable and speculative.
Not everything should be defined
Some notions are useless and futile
Not every question has an answer
Some questions are not worth asking
Not every assertion can be proven
Some assertions are a waste of time***
In fact, the simplest concepts defy definition
... without leading to contradiction
... And the simplest conjectures defy proof.
*** 'Physical' time is not so much a concern for pure mathematics.
Time is regarded as a continuous real variable or a function of causality.
Thus far, pure mathematics makes no qualitative definition of "TIME"
-- but maybe in the future.
Whatever time is, you don't have
much of it to waste...
One could say, 'That's life!" ... but it really isn't - it's mathematics
back to the beginning
In Reality,
Maybe Mathematics is a type of Science Fiction,
Awe
Inspired by
The Fall of an Arrow;
The Moon and the Stars above;
The Knowledge of Seasons and Fear of Hunger;
Charting the Seas and Mountains in Dread of being lost;
The Need to destroy Enemies who would that you be destroyed;
The Hope of seeing into the Center of the Spirit of Luck and good Fortune;
The Euphoria experienced when struck with Insight into the Workings of the Universe.
Galileo asserted that The Creator was a mathematician
But perhaps God created mathematicians
so there would be Men and Women who could understand His work
if He lost interest in the Universe and decided to walk.
Or evolutionary pressures created mathematicians accidentally
by selecting for creatures more fit to survive
in a universe that favored cleverness
Either way, your ancestors survived to dance for their DNA
and to breed timely young before they themselves eventually died
That you might be able to learn the talk & walk the walk
'Til it begins to make sense even as the lights go dim.
Sets Begetting Sets
Given any two sets, A and B, the cross product, AxB is the set of all ordered pairs of elements,
(a, b) in which the first member of the pair is in set A and the second member, b is in set B
AxB = {(a,b): a A, b B}
Set A is a subset of B if and only if all the elements of A are also elements of B:
A B <=> for every
a A, then a B
Given the set A, the Power Set, A! is the set containing all subsets of A.
-- but don't stop there, the Power set of a Power set is even bigger.
Identity between sets Given two sets A and B,
if you can demonstrate that A B and A B,
then A = B
This is the most basic way to establish identity between sets
-- later we discuss other concepts of equality based on mappings
Nested sets Like the Matryoshka or Russian nested dolls in which each doll fits into the next doll,
a series of subsets can be put in order and nested - however, if two subsets each have even
one element that is not also a member of the other set, then they cannot be nested.
A B B
...the empty set is a subset of every set and every set is a subset of itself...
Nested sets are not just for the birds
Mappings from one set to another Given two sets A and B, a mapping (or function) f is a correspondence between each element in A and
a single element in B.
This is expressed as
f:
A B
a f(a) = b
where f(a) B for all a A
one can consider a mapping f to be a subset of the cross product, AxB of sets A and B.
a one to one mapping, from A into B assigns for each element in set A
precisely ONE unique element in B
Cardinality of sets Some sets are 'bigger' than others.
the term "Cardinality" refers to the number of elements contained in a set.
If there exists a one-to-one mapping from a set A into set B, then the cardinal number of A
is no greater than the cardinal number of B.
If there also exists a one-to-one mapping from B into A, then the cardinality of A and B are equal.
this is written as /A/ = /B/
this is analogous to the numerical idea that a = b if a b
and a b
Counting with natural numbers can never exceed the finite
Inevitably this leads to the concept of infinite.
Still, an infinite set can be countable if there is a one-to-one mapping from that set into the set
of Natural Numbers, N
Coyoteman's INFINITY IN A FINITE EXISTENCE PARADOX
a) Your conception began as the fusion of two haploid gametes, an ovum and a spermatozoa,
which then began to divide exponentially in your embryonic phase.
Thus your primordial anlage survived passage through a transient portal of life
as a fragile, single-celled, 20 micron zygote. you, the once upon a time zygote were not a variety
of ruminating mammal with cloven hoof whose eyes have oval pupils.
That lucky little cell had 48 chromosomes composed of four base pairs of DNA
in a single nucleus with a cytoplasm containing a finite number
of your mother's mitochondria and other organelles. Each cellular component had a
finite number of molecules, each containing a finite number of atoms, each of which
was composed of a finite number of elementary particles in a finite quantity
of quantized, statistically predictable physical states.
Your entire existence was finite from the moment of conception.
You ended up with a trillion brain cells alone,
and trillions of other cells, some of which retain the ability to divide 50 times over
your lifetime. All told, you will be represented by a finite quantity of cells over
your lifetime, each of which have a finite number of fundamental particles.
Further, over your short lifetime, each fundamental particle can occupy only certain
discrete, quantized energy states and 'positions'.
Your entire lifetime, the sum total of your lifelong existence is simply finite.
Finite though you are, you can still conceive of the infinite,
generate infinite sets, and analyze infinite sequences and series.
Makes one think, eh?
b) Speaking of thinking, consider the maximum amount of information
your central nervous system can handle over a lifetime.
There are four types of neurons in the CNS: neurons; interneurons; sensory neurons;
and motor neurons.
For the purposes of calculation, we will regard all four kinds of neurons equally.
The action potential, AP is the signal used to transmit information
efficiently and precisely
from one neuron to another neuron at a distance.
The fundamental unit of information exchange in the nervous system or
'neural code'
is represented in the frequency and pattern of action potentials.
The average AP has a 2 msec duration.
The absolute refractory period is the interval beginning with the rise of the AP
voltage spike and
extending into fall of the spike during which a minimal recovery period occurs.
Once an AP is initiated, only after this absolute refractory period (about 1 msec)
is it possible
for a subsequent AP to take place. Thus, the maximum firing rate of action
potentials for a neuron is given by:
frequency = 1/(1msec) = 1000 impulses/sec.
Tune your dial to < 1000 Hz on the EEG band
for the broadcasting frequency of wCNS
Central Nervous System Broadcasting Corporation
back to the beginning
If you have a trillion neurons, and the maximal firing rate is 1000 impulses/sec,
and you live 100 years, then the total number of central nervous system nerve impulses
you might enjoy in a long lifetime would be
TOTAL NEURAL IMPULSES =(1000 impulses per neuron /sec)(31,536,000sec/year)(100years)(10 exp 12 neurons)
or more succinctly stated:
MAXIMUM NUMBER OF NEURAL IMPULSES
IN THE HUMAN BRAIN OVER 100 YEARS
= 3.16 x 10 exp 24
That would be a bit more than 3 followed by 24 zeros.
It is reasonable to remember this as approximately equal to x 10 exp 24.
Let's illustrate just how big a number the Thought Limit is compared to other common physical processes:
Avogadro's number = 6.023 x 10 exp 23 represents 1 mole,
i.e., the number of molecules in a molar volume (22.414 liters) of an ideal gas
at standard temperature and pressure
(STP conditions => 1 atmosphere pressure, 273.15 deg Kelvin).
Proportionally, the maximum total number of neural impulses you could possibly have in a hundred years
is roughly equivalent to the number of gaseous molecules contained in 117.52 liters at
STP conditions
- this volume is roughly equivalent to a spherical balloon inflated with air
to a diameter of 60.78 cm (just under 2 feet)
or nearly enough to fill a cubic box 49 cm (just less than 20inches) on each side - so how large was Pandora's Box and what worse than really bad belief systems
and vile, intolerant self-righteousness could have filled it?
That is not a big slice of space but let's play with the idea a bit: imagine that the lid on the box is opened
perhaps the dissipation of the vapors of your neurological existence is analogous to death.
In the balloon scenario,
if the sphere floats away, maybe it would represent an intellectual lightweight.
If the sphere sinks to the floor, perhaps it is 'too heavy' suggesting a life of sadness.
If you rupture the balloon with a spark and it explodes, then perhaps that would represent
an inflammatory character or a firebrand...
in any case, your thinking days are over...
Since all thought is based on coordinated volleys of neural impulses from
collections of neurons,
this number represents the absolute THOUGHT LIMIT over a lifetime.
Thus, the neurological consequences of your life's experience: of any perception or pleasure,
initiated action, intuition, idea or insight, thought or theory, feeling or fantasy;
All of these mental processes are capped by the THOUGHT LIMIT, and then you die.
So, what do you think?
c) Even the physical universe that we inhabit may be finite but unbounded.
Given the quantum nature of light and matter, one could argue that all the elementary particles,
their energy states and even momentums total a finite number.
This is not to say that a person could determine these precisely and completely.
Nonetheless if so, then
in the entire physical universe there is nothing infinite.
Yet, the human mind has created infinitely many concepts (e.g., numbers) and infinity itself.
What is the number of elementary particles in the universe?
All those infinities are just in your head
Beyond the reach of ordinates, when natural measure is taken to limits,
it becomes apparent that some sets are infinite. Some sets are countable
- i.e., there exists a one to one correspondence between each element
and one of the natural numbers.
Some infinite sets are bigger than others
- the number of elements in an infinite set determines cardinality.
the cardinality of the set of Natural Numbers, N
is called "aleph null", symbolized by 0
Some infinite sets are un-countable
-hopelessly beyond enumeration by the Natural Numbers, N.
There are infinitely more real numbers even in the subset of
between 0 and 1 than all of the Natural Numbers = {1, 2, 3, ...}.
If you don't understand, just take it as a given...
Some infinite sets therefore have greater cardinality than other infinite sets.
There are an infinite number of cardinalities,
an infinite number of 'degrees' of infinity
Counting up cardinalities with Hebraic characters as first used by Georg Cantor:
aleph null
= 0
aleph 1
= 1
aleph 2
= 2
...
and on without end...
So the Natural Numbers, N are the spine to the body Mathematics
But all other numbers are not necessarily 'unnatural'.
Zero is still the greatest discovery since fire.
Integers Z = {0, 1, -1, 2, -2,...} were invented by negative thinking.
Rational numbers Q = {p/q: p,qZ, q not equal to 0} can drive children crazy
Dividing by zero is verboten and undefined
Though taken to the Limit, division by zero becomes approachable...
The symbol Z for "die Zahlen" was first used by German mathematician
Edmund Landau (1877-1935) around 1930. N. Bourbaki and the 'French group' formalized this convention as well as
the use of the symbol Q for "der Quotient" during the 1930s.
The Social Behaviour of elements in a Set…
The elements of a set can be combined in ordered pairs like crossed fingers
And mapped back into the same set like light beams falling back into a black hole
However, such operations (like adding, multiplying, etc.) must map each pair
to a single, specific, target element
Like society, sets may be structured with strictly observed AXIOMATIC Rules and Regulations
Axioms specify the ways combined elements behave
When combined by operations mapping pairs of elements… This imbues a set with a definite intrinsic structure
As with integer arithmetic, groups, rings, algebras, fields, vector spaces…
Or if there are spatial properties, sets can be geometric or topological
As diverse as politics but much more predictable
View my movie, "Coyoteman IsoMorph" and see what happens to me when the sun goes down...
This will load just to the right though it might take a few minutes...
Match up key features such as the eyes, ears, nose and teeth, and you can 'morph' the image of a man's face to
the face of a coyote.
Similarly, one set can be 'morphed'into another set by mapping corresponding points
appropriately so that intrinsic structure is preserved.
Suppose you have two sets, A and B, each with a defined corresponding operation between elements that obeys
the same list of rules or axioms; Then these two sets are in a sense EQUIVALENT if mapped elements in B behave 'the same way' as
unmapped elements in A. Such a correspondence between the sets that respects the pattern of behaviour of combined elements
in each set reflects
similar or equivalent intrinsic structure between sets.
In the case of
"Coyoteman IsoMorph", set A consists of the set of points on the man's face and set B is the set
of all points on the surface of the Coyote's face.
DEFINITION: If two sets, A and B both have a defined operation
*
that follows the same axiomatic rules of combination,
then a mapping f: A B
that is a one-to-one correspondence between the sets
is called an "isomorphism" if for h, k A,
f(h*k) = f(h)
*f(k)
Isomorphisms provide a fundamental means for defining EQUALITY between different sets
based on "equal or corresponding" features within the intrinsic gestalt of both sets.
There are significant distinctions between Alge bra and Wonder bra
Wonder bra deals with duality
Two symmetrical conical 3-dimensional volumes
Understanding the Wonder bra requires hands-on experience.
Algebra deals with how elements cavort and spawn within a set.
Never ask exactly what an element is because I won't answer.
"element" or "member" is another 'primitive', undefined term
Operations between elements is sterile but non-medical
Adding friends and subtracting enemies
is one operation for elements in the slow lane.
Multiplying like rabbits is a fast track operation
by which elements make products.
All algebraic operations must be finished by dawn
Or soon thereafter…
They always follow their rules and axioms,
But there are never more than a finite number of operations in Algebra
Similarly, there is only so much you can do in a Wonder bra…
Nonetheless, out of the womb of algebra come IRRATIONAL NUMBERS:
The irrational numbers can be conceptualized
by Dedekind Cuts, open knife wounds bleeding irrationally from nicks in the Set of Rational Numbers,Q.
Like the Real Numbers R, irrational numbers R are uncountable and hopelessly
outnumber the rational numbers.
Just like people
Irrational numbers that arise as roots of polynomials with rational coefficients
are called 'Algebraic numbers'.
If an integer is not already the nth power of some other integer, then the nth root must be irrational.
This is easily proven based on the following argument (one of my teenaged
brainstorms): for any integer z, assume there exist
p,qQ such that (p/q)^n = z.
For z to be an integer, q^n must divide p^n. However, given the decomposition of each integer into unique prime
divisors, it is apparent that p must divide q - i.e., p/q is an integer already Q.E.D.
The Fundamental theorem of Arithmetic stating that every natural number can be uniquely represented as the
product of prime numbers (e.g., 12 = 2x2x3). This was rigorously proven in 1801 by Karl Friedrich Gauss (1777-1855).
Intuitively, it is easy to see that the cardinality of the set of algebraic numbers
is 0, the same as the
natural numbers, N.
To get beyond Algebra, you have to get down and get analytical
But get too analytical in other circumstances
and you probably won't even get past the Wonder bra...
---Don't even worry statistically about the exact odds or that event…
Transcendental numbers (e.g., and e) are beyond
the reach of algebra and finite reckonings
...non-algebraic irrational real numbers, never to serve as the roots of polynomials,
transcend all of algebra.
back to the beginningThe Continuum Hypothesis
Georg Cantor argued that the cardinality of the real numbers, R is the least cardinal
number greater than that of the natural numbers. However, this assertion, known as the
"Continuum Hypothesis" has never been proven.
Toward the end of his career, the founder of modern set theory,
Georg F.L.P. Cantor (1845-1918) was already aware that arguments involving the use of
"too large sets" could lead to contradictions. Intuitively, the paradoxes were similar to
the problem posed by the questions:
"If God is truely omnipotent, can he, she or it create
a stone that is too big for him, her or it to move?
In other words, can an all powerful God also be perfect? Can God create a problem that is for Godself, insolvable?
By 1905, Henri Poincare' and Bertrand Russell
suggested that the paradoxes were due to a characteristic 'circuity of definition', referred to
as impredicative definitions. .
If a set T and an element t are so defined that t T,
but is defined only by reference to T,
then the definition of T, or of t is said to be impredicative.
Axiomatic set theory arose from efforts to reformulate set theory to exclude "too
large sets". The first system of axiomatic set theory was developed by Ernst Zermelo
in 1908.
This was later improved by Adolf Fraenkel and Thoralf Skolem in 1922-23.
Currently, standard
axiomatic set theory, referred to as ZFC probably provides the most basic and workable system
for mathematics from which such paradoxes apparently cannot be deduced.
In 1938, Kurt Godel demonstrated that Cantor's Continuum Hypothesis can be
added as a new axiom to ZFC set theory without introducing a contradiction.
Thus, the foundations of mathematics provide substance for a "Continuum of Debate and
Investigation"...
A subset of the real numbers, R, from which a countable subset was removed would still have the
cardinality of R. In fact, the cardinality of all the real numbers has the same cardinality of
the open interval (0,1) = { x R: 0 < x < 1 } as shown by the following
one-to-one correspondence, mapping f: R → (0, 1):
1 / (x + 2)
for x > 0
which decreases through a
range = ( 0, .5 ) as x increases
1 / 2
for x = 0
to leave our curves connected
1 + [1 / (x - 2)]
for x < 0
which increases through a
range = ( .5, 1) as x decreases
This interval is a 'continuous' subset of the real numbers that contains no integers.
This simple example demonstrates how removing a countable set,
Z from the R does not reduce the cardinality of R.
Similarly, it is intuitively plausible that the cardinality of the non-algebraic reals is the same
as the cardinality of the reals, given that we are dealing with the set of reals excluding the
countable set of algebraic numbers.
Corollary:
Were the Continuum hypothesis true, then one could use it to argue that
the cardinality of the set of non-algebraic irrational numbers
would be aleph-1
(
1),
as would be the cardinality of the real numbers, R.
If a notion seems plausible enough to lead to theory,
in the end, theory should correspond
to the intuitive ideas that motivate generalizations...
One can encircle the square, but one can NEVER square the circle
and 'circling the square' is really the same problem...
All these numbers are REAL and I suppose, eternal...
Anyway you cut it, the cardinalities of the Real Numbers, /R/, and of
1 (if that is indeed, different),
are infinitely greater
than the cardinality of the Natural Numbers, /N/
= 0
- now you really know something of reality...
Nonetheless, even pure Imaginary numbers i =
seem real enough to me. The symbol ' i ' was first used by Swiss mathematician,
Leonid Euler (1707-1783; see portrait below)
though the concept was 'in the air' and showed up in the work of various mathematicians.
Just treat i as an algebraic variable with the property that:
= -1.
a most productive invention!
Men erroneously believe that imaginary numbers can drive women crazy…
-- it's not the number but the reckoner who counts…
Complex numbers, C = {a + i b: a, b R } are really pretty simple.
Using complex coefficients, you can solve any polynomial
equation with complex roots though you have to use non-algebraic ideas to prove it.
Analysis pushes finite numbers of operations to the limit.
Ditto for Complex analysis.
The symbol "C" for the set of complex numbers has been standard at least since the
Survey of Modern Algebra 2nd Edition by Birkhoff and MacLane - one of my courses at Purdue University in 'LayFlat',
Indiana used a later edition of
this classic text.
Though I had not spoken to her for a week, a girl once snapped at me,
"I thought I told you to give me some space!"
I realized abruptly that the 'initial values' for
the parameters of the relationship were no longer pertinent.
If temperature and pressure are (physical) state dependent functions,
(i.e., dependent only on
present conditions without reference for how a system arrived at that state),
then certainly, love is an emotional 'state dependent function'.
It doesn't matter where things have been. It matters only where things are...
Space became our final frontier - probably more than a five year mission to explore…
Nonetheless, I looked for an angle from which to work through this…
it became clear at once that an angle is not really a matter of degree.
In fact, the most important thing about an angle is not the sharp little point
But rather the arc it cuts on a unit circle centered at that sharp little point.
Multiply two complex numbers and you add their angles, so go figure.
= cos() + isin()
it follows that
* = [cos() +
isin()] * [cos() + isin( )]
and
* = = cos( +
) + isin( + )
No one should have to memorize all those half-angle and sum of angles formulae for trigonometry
- nor is there much need for those intuitive, graphic geometrical derivations of such equations.
Just take the first equation above and use it as a squeeze box for all those mathematical 'etudes'.
Euler was to mathematics
As Nostradamus was to prophesy
If the Complex plane is a bed with satin sheets,
then Complex Analysis is the sexiest of mathematical foreplay
The Unity tree:
Solutions of the equation = 1 are given by
evaluating
e exp (2i m /n) where m = 0, 1, 2, ... , n-1.
For instance, to find the 5th roots of 1 (i.e., solutions of the equation
x ^ 5 = 1), one can consider the following values using the fact that
2 radians = 360 degrees:
One then needs to get the values of cos and sin for angles in multiples of
360 / 5 degrees which includes values contained in the set {72, 144, 216, 288, 360}
degrees, d 72 144 216 288 360
X = COS(d) .3090 -.8090 .5878 .3090 1
Y = SIN(d) .9511 .5878 -.8090 -.9511 0
Root = X + i Y .3090 + i .9511
-.8090 + i .5878
-.5878 - i .8090
-.9511 + i .3090
1
QUADRANT I II III IV I (x-axis)
Each complex number, X + i Y in the Root column equals unity (1) when multiplied by itself five times...
Note that if one continues to take multiples of 72 degrees, the same points on the perimeter of the
circle are repeated - in other words, these are the five unique roots of the equation, R ^ 5 - 1 = 0.
This is of course, an example of the Fundamental Theorem of Algebra, first proven by a young Karl F. Gauss...
The number 'one' has roots in the complex plane
and gives shade to the algebraist who is not allergic to the transcendental number,
e = 2.718281828..., a number named after Leonid Euler
This transcendental is God's Telephone Number (in base 10) to 9 digits
...Trouble is, you need his office extension,
i.e., the rest of the number precisely to reach him at work...
We conclude with
ELEMENTS OF THE SET OF ALL IDEAS IN THE UNIVERSE
Fanciful notions and generalizations:
The astral plane and the complex plane are separated between two groups of people:
...Those who read easy books about fantasy and those who read really hard books about fantasy.
Mathematics is esoteric but not occult -- maybe God is clever but not a sadist!
For the Queen of science, mathematics really sleeps around a lot.
Scattered points of mathematical history:
-Kurt Godel's funny little joke: you can prove that some things can't be proven
In the twentieth century, the abstraction of logic
quit making sense to the average person.
-John von Neumann was THE party animal of the
Princeton Institute for Advanced Studies
but they didn't have a radio show....Rush Limbaugh was not a participant.
Some of mathematics is improbable, some is imprecise and some is chaotic
But all of it is accurate.
-If Newton was the Bach of mathematics, then Gauss was the Beethoven.
-If Newton was the da Vinci of mathematics, then Gauss was the Picasso.
-f Newton was the Michelangelo of sculpture, then Gauss was the Rodin.
-If Newton was the Shakespeare of mathematics, then Gauss was the Goethe.
-If Newton was the Alexander of military science, then Gauss was the Caesar.
-If Newton was the Michael Jordan of basketball, then Gauss was the Charles Barkley?
Do you think anyone will be talking about Michael Jordan and Charles "I am not
a role model" Barkley in 100 years?
......I don't think so!......
To carry this just a bit beyond too far already:
If Galileo were the John the Baptist of Mathematics,
Would Newton be The Messiah of Mathematics
***
And perhaps Karl Friedrich Gauss the Pope of Mathematics?
***After all, Isaac Newton was born in
Woolthorpe, Lincolnshire, England on Christmas Day, December 25, 1642. However, he did not die on Easter (April
18th) but rather on March 20, 1721. Sir Isaac was buried one week later in Westminister Abbey.
A victim of gravity, he 'fell' ill and thus far, has remained down for much longer than three days...
With regard to all these DWEMs (dead white European males), Middle Eastern Jewish founders of Christianity
and a couple of African American basketball players
-- that's all I have to say about that...
Naturally, I am against cloning humans. However, what if we could exhume Newton or Gauss or a few notable people
of that caliber and clone them just to see what would happen...
But what if these great intellects were principally products of their lucky day and age
and cast in another historic period would have been just above average to outstanding?
What if their clones turned out to be complete disappointments?
Imagine if one were to clone Henry Ford or John D. Rockefeller with the idea that the clone would make
a fabulous fortune for you? I would expect that these particular fellows would have
an excellent chance of achieving upper middle class standing if they worked really hard…
Only a few can be at the right place at the right time.
Would Thomas Jefferson be bored these days
with no revolution to foment, no government to found and no new society to guide toward greatness?
Having managed to enjoy and fulfill his life in his own day when illiteracy and provinciality
were the standard, I do think a man of Jefferson's intellectual capacity could keep busy today despite
the extremes of insipid consummerism and unquestioning fundamentalism.
With such extraordinary verbal and social gifts, one shouldn't be surprised were our neo-Thomas
drawn anew to a legal career. I could imagine the 21st century Thomas Jefferson clone
on late night television, the product now of a bicentennial nation, making a pitch about representing you
if you have experienced personal injury in an accident.
He might promise to get you the money you deserve and emphasize that he "would get you your money!" and
how he "goes crazy just thinkin' about the money!"
A man of that depth of passion, general intelligence and ability to manipulate would undoubtedly develop
a large and successful law practice and perhaps a well-known law firm, as long as his brains were not
sucked dry by passively watching television and playing Gameboy.
As an experiment, it would be fascinating to see if today's neo-Thomas pursued a career as an attorney at law,
seeking to protect "down trodden masses" and glorify the common man -- if so, then one might conclude that
his Democratic leanings were personality traits, hard-wired in his aristocratic, Anglo-Saxon brain. It
might simply depend on whether the young and impressionable neo-Thomas read the anti-totaliarian (Objectivism)
philosophy of Ayn Rand rather than the Empiricism and English Liberalism of the egalitarian John Locke.
What might the John D. Rockefeller clone attempt? Imagine an American company whose start up
was funded by an over-extended private collection of Visa and Master cards followed by rapid growth
powered by going public with common stock on the NASDAQ
-- for instance, this company, call it Tongue In Cheek International, (TICI) could produce paranephelia for those who follow the Islamic religion.
Besides prayer rugs and a gold leaf Koran in large print, they could market portraits
of Mohammed and his family or depictions of Allah looking down from the heavens at Mohammed -- you think?
These could be produced by cheap labor in China and distributed by European and
Israeli companies on contract. Maybe that would help us overcome this infidel image
of Westerners in the Moslem world.
Assuming initial success, product diversification could include
Running shoes for al Qaeda,
Textbooks on Algebra - there were truely great algebraists in the Middle East up to the 8th Century A.D. Books on Truth and Body Language for Middle Eastern potentates,
A shuttle service to Mars to give us all a bit more space
- Then we might actually be able to get along together separately.
There is so much more to tell yet too much more to tell in a day.
Mathematics provides the 'shorthand' with which short-lived, 70 Kg
creatures as we can
understand (to a degree) something so vast as the physical universe
in the small space and workings of a human mind
14 billion years of existence and eternity to follow
collapsed as a lattice of events within the timeline and intellect of a human life
That is the most extraordinary twist in the riddle of existence
A concluding comment, a lament actually:
Why didn't all these movies glorifying mathematicians (e.g.: Good Will Hunting, A Beautiful Mind, Pi, etc.)
come out earlier on when it might have helped me with the 'chics'? Some might argue that it wouldn't have helped anyway
The Hawk molts. Feathers fall haphazardly, sailing to the Ground.
The Moon hides all Thought in her Veil of Shadows - furtive Contemplation,
A cool Radiance just beyond the curved Horizon, just beyond View.
Coyote, wandering alien, all charms so strangely cast off, loping, casts about.
Coyote, restless magical Fool, as if in pursuit of an idea nearly forgotten,
Searches the dark Night for that faint vanishing Scent of Dreams gone by.
Robert A. Hendrix, (c)February 20, 2003
All images are copyrighted. Images were scanned from my personal collection of
historic lithographs with the exception of the caricature of John von Neumann which was
created by Andy Hendrix, the Deutsche Bundespost stamp of Karl Friedrich Gauss, and the 1999 US Post fdc
featuring Ayn Rand.
Robert A. Hendrix, MD, longtime sets maniac...
Copyright March 2002Return to top
Regarding the MIDI file you should be hearing: Erika's Song This is one of the few pieces of music I have ever composed specifically
for a female whom I loved. I wrote this short piece in 1988 on the occasion of my daughter's first
birthday. It was performed several times as a prelude for Sunday services at Ardmore Methodist Church near
Philadelphia. Recently, I found the score in a drawer -- after correcting a few mistakes in transcription, I scored it with
slight revisions using a demo version of Noteworthy Composer which I had downloaded. This software is fabulous!
It is a very intuitive, user-friendly musical score/note & staff processor and it allows one to save the file
in a MIDI format. I am pleased to have it now in a copyrighted (all rights reserved, etc.) form to play on the opening of this webpage.
I hope you enjoy "Erika's Song" by Robert Coyoteman Hendrix
DEDICATION: Dr. William Stephen Piper (1940 - ),
Stanford University alumnus, co-author of Basic Abstract Algebra (Otto F.G.
Schilling, W. Stephen Piper, Allyn & Bacon Publishing Co, 1975), faculty
member at Purdue University before he moved on to pursue Operations Research in the
Washington, D.C. area. Dr. Piper was my mentor when I was at Purdue University.
An ideal role model as a true gentleman and scholar, he respected my
ambitions and eccentricities and tolerated my foibles and inconsistencies.
Dr. Piper was the most effective and inspiring teacher I ever had.
I wish I had thanked him before now.
Other sources and websites of interest on the History of Mathematics and Set Theory
A Survey of Modern Algebra (Akp Classics) by Saunders Mac Lane, Garrett D. Birkhoff ,
Published by A K Peters Ltd; 5th edition (January 1997); ISBN: 1568810687
Men of Mathematics by Eric Temple Bell; Publisher: Touchstone Books; ; Reissue edition (October 1986); ISBN: 0671628186
Comment: never mind the 'political correctness' of the last title.
Bell authored the work long before everyone became so nervous and touchy about gender typing and every other damned thing...
It is a very worthwhile read for any Homo sapiens
-- after all, isn't sapientia the Latin word for wisdom and good sense?
Aren't we the 'wise' hominids? Perhaps one should not try to 'define' oneself too rigorously lest it falls into foolishness?
Wouldn't it be wiser to think of oneself as a 'primitive term' to avoid 'inconsistency', contradiction and even hypocrisy?
in a Finite Existence Paradox
Robert Hendrix & Mathematics From the View of the Hawk
= { }.
{n}
(caricature by Andy Hendrix)
DEFINING EVERY SINGLE CONCEPT (too rigorously)
Galileo asserted that The Creator was a mathematician
A, b 
B <=> for every
a 
B,
then A = B
B 
f(a) = b 
b
and a 
b
x 10 exp 24.
0
seem real enough to me. The symbol ' i ' was first used by Swiss mathematician,
Leonid Euler (1707-1783; see portrait below)
though the concept was 'in the air' and showed up in the work of various mathematicians.
Just treat i as an algebraic variable with the property that:
= -1.
= cos(
) + isin(
= [cos(
) + isin(
= cos(
= 1 are given by
evaluating
e exp (2i 
As an experiment, it would be fascinating to see if today's neo-Thomas pursued a career as an attorney at law,
seeking to protect "down trodden masses" and glorify the common man -- if so, then one might conclude that
his Democratic leanings were personality traits, hard-wired in his aristocratic, Anglo-Saxon brain. It
might simply depend on whether the young and impressionable neo-Thomas read the anti-totaliarian (Objectivism)
philosophy of Ayn Rand rather than the Empiricism and English Liberalism of the egalitarian 
