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# The Reasonable Ineffectiveness of Mathematics [Point of View]

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## Abstract

The nature of the relationship between mathematics and the physical world has been a source of debate since the era of the Pythagoreans. A school of thought, reflecting the ideas of Plato, is that mathematics has its own existence. Flowing from this position is the notion that mathematical forms underpin the physical universe and are out there waiting to be discovered. The opposing viewpoint is that mathematical forms are objects of our human imagination and we make them up as we go along, tailoring them to describe reality. In 1921, this view led Einstein to wonder, "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" [1 ]. In 1959, Eugene Wigner coined the phrase "the unreasonable effectiveness of mathematics" to describe this "miracle," conceding that it was something he could not fathom [2]. The mathematician Richard W. Hamming, whose work has been profoundly influential in the areas of computer science and electronic engineering, revisited this very question in 1980 [3].
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P 0 I N T 0 F V I E W )
The
Reasonable Ineffectiveness
of
Mathematics
By
DEREK
ABBOTT
School
of
Electrical and Electronic Engineering
The University
of
The na
tur
e
of
th
e relationship be
tw
een
ma
th
ema
tics a
nd
th
e
physical
worl
d has been a
so
ur
ce of debate since
th
e era
of
th
e
Pythagoreans. A school of
th
oug
ht
, reflecting
th
e ideas of P
la
to, is
th
at ma
th
ematics has its own existence. Flowing from
thi
s position
is
th
e notion
th
at ma
th
ematical
fo
rm
s
und
erpin
th
e physical universe a
nd
are
o
ut
th
ere waiting to be discovered.
The
opposi
ng
viewpoi
nt
is
that
mathematical forms are objects of
our
human
imagi
nat
ion
and
we make
them
up as we go along, tail
or
i
ng
them
to
describe
re
ality.
In
1921, this vi
ew
l
ed
Einstein to wonder,
"How
can it be
that
mathemat
ics, being after all a product of human thought which is i
ndependent
of experience, is so admirably appropriate to the objects of reality?" [1 ].
In
1959, Eugene Wi
gner
coi
ned
the phrase
"the
unreasonable eff
ect
iveness
of mathematics"
to
describe this
"m
i
ra
cle," concedi
ng
that
it was something he
could
not
fathom [2]. The mathematician Richard W. Hamming, whose
work
has
been
profoundly in
fl
uent
ial in
the
areas of computer science
and
electronic
eng
i
neer
ing, revisi
ted
this very questi
on
in 1980 [3].
Di
gita
l Object
Ident
ifier: 101109/J
PROC
.2013
227
4
90
7
Hamm
i
ng
raised f
our
interesti
ng
propositions
that
he
be
li
eved fe
ll
short of providing a con
cl
usive expl
a-
nat
i
on
[3].
Thus
, like Wi
gner
before
him, Hammi
ng
resi
gned
himse
lf
to
the idea
that
mathematics is unrea-
sonably effective. These f
our
poi
nts
are: 1)
we
see
what
we look for; 2)
we
sel
ect
the ki
nd
of
mathemat
ics
we
look for;
3)
comparatively f
ew
problems;
and
4)
the
evol
ut
i
on
of
man
provi
ded
the
mod
el.
In
this
art
icle, we will
quest
i
on
the
presupposition
that
mathemat
ics is as
effective as
cl
aimed
and
thus
remove
the quandary of Wi
gner's
"m
i
ra
cle,"
st
viewpoint.1
We
will also revisit
Hamm
ing's f
our
propositions
and
show
how
they may
indeed largely explain
that
there
is
no
mira
cl
e, given a reduced level of
mathematical effectiveness.
The
or
a
mo
-
ment
of indulgence, where we will
push these ideas to the extreme, ex-
tend
ing
them
to all physical law
and
models. Are they all truly reified? We
will question
the
ir absolute reality
and
the
question: Have we, in some
sense, generated a partly anthropo-
centr
ic physical
and
mathemat
ical
framework of
the
wo
rld around us?
Why
shou
ld
we
care?
Among
sci
ent
ists
and
engineers,
there
are
those
that
ons
and
there
are those
that
pref
er
to
"shut
up
and
calculate." We will at-
tempt
to explain
why
there
mi
ght
be a
'This
explains the
in
verted title of the
present article, "The reasonable ineffectiveness
of mathemati
cs
."
0018
9219
VoL
101,
No.
10,
Oc
tober
2013
1
PRO
C
EE
D
IN
GS
OF
THE
IEEE
2147
useful payoff in resolving our philo-
sophical qualms and how this might
assist our future calculations.
I. MATHEMATICIANS,
PHYSICISTS, AND
ENGINEERS
The following is anecdotal and is by
no means a scientific survey. How-
ever, in my experience of interacting
with mathematicians, physicists, and
engineers, I would estimate that
to a Platonist view.
2
Physicists, on
the other hand, tend to be closeted
non-Platonists. An ensemble of phy-
sicists will often appear Platonist in
public, but when pressed in private I
can often extract a non-Platonist
confession.
Engineers by and large are openly
non-Platonist. Why is that? Focusing
on electrical and electronic engineer-
ing, as a key example, the engineer is
well acquainted with the art of ap-
proximation. An engineer is trained to
be aware of the frailty of each model
and its limits when it breaks down.
For example, we know that lumped
circuit models are only good for low
frequencies.
An engineer is also fully aware of
the artificial contrivance in many
models. For example, an equivalent
circuit only models the inputs and
outputs of a circuit, and ignores all the
internal details. Moreover, the engi-
neer knows the conditions under
which these simplifications can be
exploited.
An engineer often has control over
his or her ‘‘universe’’ in that if a sim-
ple linear model does not work, the
engineer, in many cases, can force a
widget, by design, to operate within a
restricted linear region. Thus, where
an engineer cannot approximate line-
arity, he often linearizes by fiat.
A mathematical Platonist will of-
ten argue that number is a real en-
tity, claiming that a geometric circle is
a reified construct that exists inde-
pendently of the universe. An engi-
neer, on the other hand, has no
difficulty in seeing that there is no
such thing as a perfect circle any-
where in the physical universe, and
thus is merely a useful mental
construct.
In addition to the circle, many
other ideal mathematical forms such
as delta functions, step functions, si-
nusoids, etc., are in an engineer’s
mathematical toolbox and used on a
daily basis. Like the circle, the engi-
neer sees delta functions, and for that
matter all functions, as idealities that
do not exist in the universe. Yet, they
are useful for making sufficiently ac-
curate, yet approximate, predictions.
A physicist may have nightmares
on studying a standard electronic en-
gineering text, finding the use of ne-
gative time in the theory of noncausal
filters. However, a non-Platonist en-
gineer has no qualms about such
transformations into negative spaces,
as there is no ultimate reality there.
These are all mental constructs and
are dealt with in a utilitarian way,
producing the results required for
system design.
Hamming’s paper marvels on how
complex numbers so naturally crop up
in many areas of physics and engi-
neering, urging him to feel that ‘‘God
made the universe out of complex
numbers’’ [3]. However, for the engi-
neer, the complex number is simply a
convenience for describing rotations
[7], and, of course, rotations are seen
everywhere in our physical world.
Thus, the ubiquity of complex numb-
ers is not magical at all. As pointed out
by Chappell et al. [8], Euler’s remark-
able formula ej¼1issomewhat
demystified once one realizes it mere-
ly states that a rotation by radians is
simply a reflection or multiplication
by 1.
Engineers often use interesting
mathematics in entirely nonphysical
spaces. For example, the support vec-
tor machine (SVM) approach to clas-
sifying signals involves transforming
physical data into nonphysical higher
dimensional spaces and finding the
optimal hyperplanes that separate the
data. In telecommunications, coding
theory can also exploit higher dimen-
sional spaces [9]. In both these ex-
amples, physically useful outcomes
result from entirely mental abstrac-
tions of which there are no analogs in
the physical universe.
II. DO FRACTALS HAVE
THEIR OWN EXISTENCE?
Roger Penrose, a mathematical Plato-
nist, argues that a fractal pattern is
proof of a mathematical entity having
an existence of its own [6]. It is ar-
gued that the mathematician cannot
foresee a beautiful fractal, before ap-
plying a simple iterative equation.
Therefore, a fractal pattern is not a
mental construct, but has its own
existence on a Platonic plane waiting
to be discovered.
A first objection is that there are
an infinite number of ways to display
the fractal data, and that to ‘‘see’’ a
fractal we have to anthropocentrically
display the data in the one way that
looks appealing to our senses. Per-
haps to an alien, a random pattern
based on white noise might be more
beautiful?
A second objection is that out of
an infinite number of possible itera-
tive equations, perhaps only negligi-
ble numbers of them result in fractal
patternsandevenfewerlookappeal-
ingtohumans.Taketheanalogyofa
random sequence of digits. We know
any infinite random sequence encodes
all the works of Shakespeare and all
the world’s knowledge. If we preselect
appealing parts of a random sequence,
we have in fact cheated.
At the end of the day, a given set of
rules that turns into an elegant fractal
is really no different to, say, the set of
rules that form the game of chess or
that generate an interesting cellular
automaton. The set of moves in a
gameofchessisevidentlyinteresting
and richly beautiful to us, but that
beauty is no evidence that chess itself
has a Platonic existence of its own.
Clearly, the rules of chess are purely a
contrived product of the human mind
and not intrinsic to nature.
2
The interested reader is referred to [4] for
an entertaining view of the non-Platonist
position, and [5] for a Plationist perspective.
Point of View
2148 Proceedings of the IEEE | Vol. 101, No. 10, October 2013
A Platonist will argue that math-
ematical forms follow from a set of
axioms, and thus exist independently
of our knowledge of them. This situa-
tion is no different to our lack of fore-
knowledge of a fractal pattern, before
exercising its originating equation.
What can we say of the axioms them-
selves? I argue that they are also
mental abstractions, and an example
isgiveninSectionVtoillustratethat
eventhesimplecountingofobjects
has its physical limits. Thus, axioms
based on the assumption of simple
counting are not universally real.
III. THE
INEFFECTIVENESS
OF MATHEMATICS
So far, we have argued that mathe-
matics is a merely mental abstraction
that serves useful purposes. A further
that the effectiveness of mathematics
is a ‘‘miracle’’ is to suggest that this
effectiveness might be overstated.
What we are finding in electronic
engineering is that the way we math-
ematically model and describe our
systems radically changes as we ap-
proach the nanoscale and beyond. In
the 1970s, when transistor MOSFET
lengthswereoftheorderofmicro-
meters, we were able to derive from
physical first principles elegant ana-
lytical equations that described tran-
sistor behavior, enabling us to design
working circuits. Today, we produce
deep submicrometer transistors, and
these analytical equations are no long-
er usable, as they are swamped with
too many complicated higher order
effects that can no longer be neglected
at the small scale. Thus, in practice,
we turn to empirical models that are
embedded in today’s computer simu-
lation software for circuit design.
simply fails to describe the system in
a compact form.
Another example is the use of
Maxwell’s equations for modeling in-
tegrated electromagnetic devices and
structures. In modern devices, due to
the complexity of design, we no long-
er resort to analytical calculations;
programs that use numerical methods
are now the standard approach.
The point here is that when we
carry out engineering in different
circumstances, the way we perform
mathematics changes. Often the
reality is that when analytical meth-
ods become too complex, we simply
resort to empirical models and
simulations.
The Platonist will point out that
the inverse square law for gravitation
is spectacularly accurate at predicting
the behavior of nearby planets and
distant stars across vast scales. How-
ever, is that not a self-selected case
conditioned on our human fascination
with a squared number? Furthermore,
due to inherent stochasticity in any
physical system, at the end of the day,
we can only ever experimentally ve-
rify the square law to within a certain
accuracy. While the Newtonian view
of gravitation is a spectacularly suc-
cessful model, it does not hold what
we believe to be the underlying real-
ity; it has been surpassed by the 4-D
curved spaces of general relativity,
andthisisnowthedominantview-
point until a better theory comes
along.
Note that mathematics has lesser
success in describing biological sys-
tems, and even less in describing eco-
nomic and social systems. But these
systems have come into being and are
contained within our physical uni-
verse. Could it be they are harder to
model simply because they adapt and
change on human time scales, and so
the search for useful invariant prop-
erties is more challenging? Could it be
that the inanimate universe itself is no
different, but happens to operate on a
timescale so large that in our anthro-
pocentrism we see the illusion of
invariance?
An energy-harvesting device that
is in thermal equilibrium cannot ex-
tract net energy or work from its en-
vironment. However, if we imagine
that human lifespans are now re-
duced to the timescale of one thermal
fluctuation, the device now has the
illusion of performing work. We
experience the Sun as an energy
source for our planet, partly because
itslifespanismuchlongerthanhu-
man scales. If the human lifespan
were as long as the universe itself,
perhaps our sun would appear to be
short-lived fluctuation that rapidly
brings our planet into thermal equi-
librium with itself as it ‘‘blasts’’ into a
red giant. These extreme examples
show how our anthropocentric scales
possibly affect how we model our
physical environment.
A. Hamming’s First Proposition:
We See What We Look For
Hamming suggests here that we
approach problems with a certain in-
tellectual apparatus, and, thus, we
anthropocentrically select out that
which we can apply our tools to [3].
Our focus shifts as new tools become
available. In recent years, with the
tems and mining of so-called big data,
role and large brute force computing
is used to search for the patterns we
are looking for.
B. Hamming’s Second
Proposition: We Select the Kind
of Mathematics We Look For
Here, Hamming points out that
we tailor mathematics to the problem
athand[3].Agivensetofmathemat-
ical tools for one problem does not
necessarily work for another. The his-
tory of mathematics shows a continual
development; for example, scalars
came first, then we developed vectors,
then tensors, and so on. So as fast as
mathematics falls short, we invent
new mathematics to fill the gap.
By contrast, a Platonist will argue
for the innateness of mathematics by
pointingoutthatwesometimesin-
vent useful mathematics before it is
needed. For example, Minkowski and
Riemann developed the theory of 4-D
curved spaces in the abstract, before
Einstein found it of utility for general
relativity. I argue that this innateness
is illusory, as we have cherry picked
a successful coincidence from a
Point of View
Vol. 101, No. 10, October 2013 | Proceedings of the IEEE 2149
backdrop of many more cases that are
not as fortuitous.
C. Hamming’s Third Proposition:
Few Problems
Taking into account the entire hu-
man experience, the number of ques-
tions that are tractable with science
and mathematics are only a small
fraction of all the possible questions
¨del’s theorem also set
limits on how much we can actually
prove. Mathematics can appear to
have the illusion of success if we are
preselecting the subset of problems
for which we have found a way to
apply mathematics.
A case in point is the dominance of
linear systems. Impressive progress
has been made with linear systems,
because the ability to invoke the prin-
ciple of superposition results in ele-
gant mathematical tractability. On the
other hand, developments in nonlin-
earsystemshavebeenarduousand
much less successful. If we focus our
attention on linear systems, then we
have preselected the subset of pro-
blems where mathematics is highly
successful.
3
D. Hamming’s Fourth Proposition:
The Evolution of Man Provided
the Model
A possibility is that the quest for
survival has selected those who are
able to follow chains of reasoning to
understand local reality. This implies
that the intellectual apparatus we use
is in some way already appropriate.
Hamming points out that, to some
extent, we know that we are better
adapted to analyzing the world at our
human scale, given that we appear to
have the greatest difficulties in rea-
soning about the very small scale and
the very large scale aspects of our
universe.
E. Physical Models as a
Compression of Nature
There is a fifth point we might add
to Hamming’s four propositions, and
that is that all physical laws and math-
ematical expressions of those laws
are a compression or compact repre-
sentation. They are necessarily com-
pressed due to the limitations of the
human mind. Therefore, they are
compressed in a manner suited to
the human intellect. The real world is
inherently noisy and has a stochastic
component, so physical models are
idealizations with the rough edges
removed.
Thus, when we ‘‘uncompress’’ a
set of equations, to solve a given prob-
lem, we will obtain an idealized result
that will not entirely match reality.
This can be thought of as uncom-
pressing a video that was initially
subjected to lossy compression. There
will always be lossy information leak-
age but, provided the effects we have
neglected are small, our results will be
useful.
F. Darwinian Struggle for the
Survival of Ideas
A sixth point we can add to
Hamming’s list is that Wigner’s sense
of ‘‘magic’’ can be exorcised if we see
that the class of successful mathemat-
ical models is preselected. Consider
the millions of failed models in the
minds of researchers, over the ages,
which never made it on paper because
they were wrong. We tend to publish
the ones that have survived some level
of experimental vindication. Thus,
this Darwinian selection process re-
sults in the illusion of automatic suc-
cess; our successful models are merely
selected out from many more failed
ones.
Take the analogy of a passenger on
a train, pulling the emergency stop
lever, saving the life of a person on a
railway track; this seems like a mira-
cle. However, there is no miracle once
we look at the prior that many more
people have randomly stopped trains
on other occasions saving no lives. A
genius is merely one who has a great
idea, but has the common sense to
keep quiet about his other thousand
insane thoughts.
ALIENS?
Mathematical Platonists often point
out that a hypothetical alien civiliza-
tion will most likely discover the num-
ber and put it to good use in their
alien mathematics. This is used to ar-
gue that has its own Platonic exis-
tence, given that it is ‘‘out there’’ for
any alien to independently discover.
Do aliens necessarily know num-
ber ?Doaliensevenhavethesame
view of physics?
Given the simplicity of geometric
objects such as ideal circles and
squares, an alien race may indeed
easily visualize them. However, this is
not true of all our mathematical ob-
jects, especially for those with in-
creased complexity. For example, an
alien race may never find the Man-
dlebrot set, and may not even pause to
find it interesting if found by chance.
An alien race might happily do all
its physics and engineering without
the invention of a delta function. Per-
haps the aliens have parameterized all
their physical variables in a clever
way, and if we were to compare we
wouldfindthatoneofourvariables
was surprisingly redundant.
Perhaps not all aliens have a taste
for idealizations, nor Occam’s razor.
Maybe all their physical equations are
stochastic in nature, thereby realisti-
cally modeling all physical phenome-
na with inherent noise.
One might also hypothesize a su-
perintelligent alien race with no need
forlongchainsofanalyticalmathe-
matical reasoning. Perhaps their
brains are so powerful that they jump
straight into performing vast nume-
rical simulations, based on empirical
models, in their heads. So the question
of the effectiveness of mathematics, as
we know it, has no meaning for them.
This thought experiment also illus-
trates that human mathematics serves
us to provide the necessary compres-
sion of representation required by
our limited brain power.
3
One might remark that many fundamental
processes rather successfully approximate linear
models, and this may again seem like Wigner’s
magic. However, is this not self-referential?
What we humans regard as ‘‘fundamental’’ tend
to be those things that appear linear in the first
place.
Point of View
2150 Proceedings of the IEEE |Vol.101,No.10,October2013
V. ONE BANANA, TWO
BANANA, THREE
BANANA, FOUR
I deeply share Hamming’s amazement
at the abstraction of integers for count-
ing [3]. Observing that six sheep plus
seven sheep make 13 sheep is some-
thing I do not take for granted either.
A deceptively simple example to
illustrate the limitations in the corre-
spondencebetweentheidealmathe-
matical world and reality is to dissect
the idea of simple counting. Imagine
counting a sequence of, say, bananas.
When does one banana end and the
next banana begin? We think we
know visually, but to formally define
it requires an arbitrary decision of
what minimum density of banana
molecules we must detect to say we
have no banana.
To illustrate this to its logical ex-
treme, imagine a hypothetical world
where humans are not solid but
gaseous and live in the clouds. Surely,
ifweevolvedinsuchanenvironment,
our mathematics would not so readily
encompass the integers? This relates
to Hamming’s Fourth Proposition,
where our evolution has played a role
in the mathematics we have chosen.
Consider the physical limits when
counting a very large number of bana-
nas. Imagine we want to experimen-
tally verify the 1-to-1 correspondence
between the integer number line, for
large N, with a sequence of physical
bananas. We can count bananas, but
for very large N, we need memory to
store that number and keep incre-
menting it. Any physical memory will
always be subject to bit errors and
noise, and, therefore, there are real
physical limits to counting.
An absolute physical limit is when
Nis so large that the gravitational pull
of all the bananas draws them into a
black hole.
4
Thus, the integer number
line is lacking in absolute reality.
Davies goes a step further and argues
that real numbers are also a fiction;
they cannot be reified as the universe
can store at most 10122 bits of
information [11].
VI. STRONG
NON-PLATONISM
For the purposes of this essay, we
have loosely labeled mathematical
Platonism as the position that ideal
mathematical objects exist and they
are waiting to be discovered. Simi-
larly, physical laws are also reified.
What we loosely refer to as non-
Platonism is the view that mathemat-
ics is a product of human imagination
and that all our physical laws are im-
perfect. Nature is what it is, and by
physical law we are, of course, refer-
ring to man’s compression of nature.
tain strong non-Platonism, where all
physical laws are tainted with anthro-
pocentrism and all physical models
have no real interpretative value. The
interpretive value of physics is purely
illusory. After all, a beam of light
passing through a slit knows nothing
of Fourier transforms; that is an over-
laid human construct.
Imagine 3-D particles passing
through a 2-D universe. A 2-D flat-
lander [12] can create beautiful inter-
pretations, which may even have
some predictive accuracy, regarding
these mysterious particles that ap-
pear, change size, and then disappear.
But these interpretations are to some
extent illusory and at best incomplete.
Inourworld,wearetrappedon
human length scales, human power
scales, and human time scales. We
have created clever instruments that
extend our reach, but we are hope-
lessly lacking in omnipotence.
In some cases, we knowingly build
up a set of models with imaginary in-
terpretative value purely for conve-
nience. For example, we can measure
the effective mass and drift velocity of
holes in a semiconductor, knowing
fully well that semiconductor holes
are an imaginary artifice. We exploit
them as a mental device because they
provide a shortcut to giving us predic-
tive equations with which we can
engineer devices.
John von Neumann stated all this
more succinctly: ‘‘The sciences do not
trytoexplain,theyhardlyeventryto
interpret, they mainly make models.
By a model is meant a mathematical
of certain verbal interpretations, de-
scribes observed phenomena. The
justification of such a mathematical
construct is solely and precisely that it
is expected to work’’ [13].
VII. IMMUTABILITY
Another way to see the potential
frailty of physical ‘‘laws’’ created by
man is to ask which principles in phy-
sics are sacred and immutable? I will
er.However,whenItriedthethought
experiment I was able to stretch my
imagination to permitting a violation
of everything we know. At some vast
or small scale of any set of parameters,
one can imagine breakdowns in the
laws, as we know them.
Is there anything we can hold onto
as inviolate under any circumstances?
What about Occam’s razor? I would
like to hold onto Occam’s razor as
immutable, but I fear that it too may
be embedded with anthropocentrism.
When classifying physical data, it is
known that God does not always shave
withOccamsrazor[14].Coulditbe
that, as the human brain demands a
compression of nature, Occam’s razor
is our mental tool for sifting out com-
pact representations?
VIII. APERSONALSTORY
As this is an opinion piece, it might be
pertinent to understand where my
opinions come from. I have a distinct
memory of being alone playing on the
floor, at the age of four, with a large
number of cardboard boxes strewn
across the room. I counted the boxes.
Then,Icountedthemagainandob-
tained a different number. I repeated
this a few times obtaining different
numbers. This excited me because I
thought it was magic and that boxes
were appearing and disappearing. But
the magic unfortunately disappeared
4
It is of interest to note here that Lloyd has
exploited black holes to explore the physical
limits of computation [10].
Point of View
Vol. 101, No. 10, October 2013 | Proceedings of the IEEE 2151
and I eventually kept obtaining a run
of the same number. In a few minutes
I concluded that my initial counting
was inaccurate and that there sadly
never was any magic. This was my
first self-taught lesson in experimen-
tal repeatability and the removal of
magic from science.
At both elementary school and
high school, mathematics was my fav-
orite subject, although I spent far too
cept of infinity. Taking a limit to in-
finity was something I simply got used
to, minus the desire to wildly embrace
it. I struggled with accepting negative
numbers, and raising numbers to the
power of zero seemed absurd.
5
Ire-
member a great sense of disappoint-
ment when I was told that vectors
could not be divided. Something was
not quite right, but then I could not
put my finger on it. After all complex
numbers contain a direction and mag-
nitude, yet can be divided. The more
mathematics I learned the more it
seemed like an artificial hodgepodge
of disparate tools, rather than a divine
order.
While I loved the beauty of math-
ematical proofs and the search for
them, it worried me that each proof
there was no heavenly recipe book.
The nature of proofs began to appear
philosophically suspect to me, for
example, how do we really know if a
proof is correct if it is too long? A
mathematical proof is the demonstra-
tion that a proposition is correct with
a level of certainty that two mathe-
maticians somewhere in the world
understand it; that was in jest, of
course, but the proof of Fermat’s last
theorem is arguably close to pushing
that boundary.
At the age of 19, in my undergrad-
uate university library, I stumbled on
a textbook that changed my life. In its
introduction, it stated that mathemat-
icsisaproductofthehumanmind.
Obviously, all my teachers must have
been mathematical Platonists, as I had
never heard such an outlandish state-
ment before. Immediately, a great
burden lifted from my shoulders, and
my conversion to non-Platonism was
instant. This was a road to Damascus
experience for me, and my philoso-
phical difficulties that haunted me
vanished.
As Hamming aptly states, ‘‘The
postulates of mathematics were not
on the stone tablets that Moses
brought down from Mt. Sinai’’ [3].
IX. WHY NOT JUST SHUT
UP AND CALCULATE?
Why should we care about the nature
of mathematics? My personal story for
one illustrates that there is greater
freedomofthought,oncewerealize
that mathematics is something we en-
tirely invent as we go along. This view
can move us ahead and free us from
an intellectual straight jacket. With
the shackles removed, we can proac-
tively manipulate, improve, and apply
mathematics at a greater rate.
If we discard the notion that math-
ematicsispasseddowntousonstone
tablets, we can be more daring with it
and move into realms previously
thought impossible. Imagine where
wecouldbenowifthecenturiesof
debate over negative numbers could
have been resolved earlier.
Another problem with mathemat-
ics today is the lack of uniformity in
the tools we use. For example, we
have the Cartesian plane and the
Argand plane. They are isomorphic
to each other, so why must we have
both? We have complex numbers and
quaternions. We have scalars, vectors,
and tensors. Then, we have rather
clunky dot and cross products, where
the cross product does not generalize
to higher dimensions.
It turns out to be something of a
historical accident that the vector
notation, with dot and cross products,
waspromotedbyGibbsandHeavi-
side, giving us a rather mixed bag of
different mathematical objects.
Clifford’s geometric algebra on the
otherhand,unifiesallthesemathe-
matical forms [8], [15] [17]. It uses
Cartestian axes and replaces complex
numbers, quaternions, scalars, vec-
tors, and tensors all with one mathe-
matical object called the multivector.
Dot and cross products are replaced
with one single operation called the
geometric product.Thisnewtypeof
product is elegant and follows the
elementary rules for multiplying out
brackets, with the extra rule that ele-
ments do not commute. You cannot
vectors do not have this restriction.
All the properties naturally extend to
higher dimensions, and thus the limi-
tations of the cross product are over-
come. This formalism is therefore
simple and powerful, and delivers im-
proved mathematical compression tai-
lored for the limited human mind.
While this approach has existed
since 1873, it has been largely side-
lined, as Gibbs and Heaviside favored
dot and cross products. However, in
physics, engineering, and computer
science there is an emerging interest
in reviving this mathematics due to its
power and simplicity. To this end, we
foreshadow a tutorial paper on geo-
metric algebra for electrical and elec-
tronic engineers to be published in
the Proceedings of the IEEE at a
later date [18].
X. CONCLUSION
Science is a modern form of alchemy
that produces wealth by producing the
understanding for enabling valuable
products from base ingredients. Sci-
ence is merely functional alchemy
that has had a few incorrect assump-
tions fixed, but has in its arrogance
replaced them with more insidious
ones. The real world of nature has the
uncanny habit of surprising us; it has
always proven to be a lot stranger than
we give it credit for.
Mathematicsisaproductofthe
imagination that sometimes works on
simplified models of reality. Plato-
nism is a viral form of philosophical
reductionism that breaks apart ho-
listic concepts into imaginary dual-
isms.Iarguethatliftingtheveilof
5
In retrospect, I am astonished with how
my mindset was so 16th century. I will argue
that it is the ravages of Platonism that can lock
us into that mold.
Point of View
2152 Proceedings of the IEEE |Vol.101,No.10,October2013
mathematical Platonism will acceler-
ate progress. In summation, Platonic
ideals do not exist; however, ad hoc
elegant simplifications do exist and
are of utility provided we remain
aware of their limitations.
Mathematicsisahumaninvention
for describing patterns and regulari-
ties. It follows that mathematics is
then a useful tool in describing regu-
larities we see in the universe. The
reality of the regularities and invar-
iances, which we exploit, may be a
little rubbery, but as long as they are
sufficiently rigid on the scales of in-
terest to humans, then it bestows a
sense of order. h
Acknowledgment
This paper is based on a talk the
author presented at a workshop enti-
tledTheNatureoftheLawsofPhy-
sics,December17 19,2008,Arizona
State University (ASU), Phoenix, AZ,
USA. The author would like to thank
all the attendees that provided useful
Platonism including: S. Aaronson of
the Massachusetts Institute of Tech-
nology (MIT, Cambridge, MA, USA);
P. C. W. Davies of ASU; G. F. R. Ellis
of the University of Cape Town (Cape
Town,SouthAfrica);G.J.Chaitin
of IBM (Armonk, NY, USA); A. J.
Leggett of the University of Illinois
at Urbana Champaign (UIUC,
Urbana, IL, USA); N. D. Mermin
of Cornell University (Ithaca, NY,
USA); L.SusskindofStanfordUni-
versity (Stanford, CA, USA); and
S. Weinstein of the University of
Theauthorisalsogratefulforanum-
ber of formative discussions on the
topic, over the years, with C. Shalizi
of Carnegie Mellon University (CMU,
Pittsburgh,PA,USA)andP.C.W.
Davies of ASU. A special thanks goes
to A. J. Leggett of UIUC, for pointing
out the problem of integer counting
in the case of hypothetical gaseous
beingsVTony was, in turn, inspired
by Hawkins [19]. The author would
also like to thank K. Wiesenfeld of
Georgia Tech (Atlanta, GA, USA), for
his thought experiment of an alien
with computational mental powers
sufficient to simulate the environ-
ment. The author asked a number of
tions to be brutal. He is grateful to
R.E.Bogner,J.M.Chappell,P.C.W.
Davies,B.R.Davis,G.F.R.Ellis,
M. D. McDonnell, A. P. Flitney,
C.Mortensen,andW.F.Pickardfor
supplying gracious brutality in the
spirit of debate.
REFERENCES
[1] A. Einstein, Geometrie und Erfahrung.
Berlin, Germany: Springer-Verlag, 1921.
[2] E.P.Wigner,‘Theunreasonable
effectiveness of mathematics in the natural
sciences,’’ Commun. Pure Appl. Math.,
vol. XIII, pp. 1–14, 1960.
[3] R. W. Hamming, ‘‘The unreasonable
effectiveness of mathematics.,’’ Amer. Math.
Monthly,vol.87,no.2,pp.8190,1980.
[4] D.C.Stove,The Plato Cult and Other
Philosophical Follies.Oxford,U.K.:
Blackwell, 1991.
[5] S.C.Lovatt,New Skins for Old Wine: Plato’s
Wisdom for Today’s World. Boca Raton, FL,
USA: Universal, 2007.
[6] R. Penrose, The Road to Reality: A Complete
GuidetotheLawsoftheUniverse.NewYork,
NY, USA: Knopf, 2004.
[7] E.O.Willoughby,‘Theoperatorjand a
demonstration that cos þjsin ¼ej,’’ in
pp. 118–119, 1965.
[8] J. M. Chappell, A. Iqbal, and D. Abbott,
Geometric Algebra: A Natural Representation
of Three-Space. [Online]. Available: http://
arxiv.org/pdf/1101.3619.pdf
[9] M.El-Hajjar,O.Alamri,J.Wang,S.Zummo,
and L. Hanzo, ‘‘Layered steered space-time
codes using multi-dimensional sphere
packing modulation,’’ IEEE Trans. Wireless
Commun.,vol.8,no.7,pp.33353340,
Jul. 2009.
[10] S. Lloyd, ‘‘Ultimate physical limits to
computation,’’ Nature, vol. 406, no. 6799,
pp. 1047–1054, 2000.
[11] P. C. W. Davies, The Goldilocks Enigma:
Why Is the Universe Just Right for Life?
London, U.K.: Penguin, 2007.
[12] E. A. Abbott, Flatland: A Romance of Many
Dimensions. London, U.K.: Seeley, 1884.
[13] J.vonNeumann,‘Methodinthephysical
sciences,’’ in The Unity of Knowledge,
L. Leery, Ed. New York, NY, USA:
Doubleday, 1955, pp. 157–164.
[14] H. Bensusan, ‘‘God doesn’t always shave with
Occam’s razor Learning when and how
to prune,’’ in Lecture Notes in Computer
Science. Berlin, Germany: Springer-Verlag,
1998,vol.1398,pp.119124.
[15] D. Hestenes, New Foundations for Classical
Mechanics: Fundamental Theories of Physics.
New York, NY, USA: Kluwer, 1999.
[16] S.Gull,A.Lasenby,andC.Doran,
‘‘Imaginary numbers are not real The
geometric algebra of spacetime,’’ Found.
Phys.,vol.23,no.9,pp.11751201,1993.
[17] M. Buchanan, ‘‘Geometric intuition,’’
Nature Phys., vol. 7, no. 6, p. 442, 2011.
[18] J.M.Chappell,S.P.Drake,L.J.Gunn,
A. Iqbal, A. Allison, and D. Abbott,
‘‘Geometric algebra for electrical and
electronic engineers,’’ in Proc. IEEE,
2014.
[19] D. Hawkins, The Language of Nature.
San Francisco, CA, USA: Freeman, 1964.
Point of View
Vol. 101, No. 10, October 2013 | Proceedings of the IEEE 2153
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