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Proving the Point: Connections Between Legal and Mathematical Proving the Point: Connections Between Legal and Mathematical
Reasoning Reasoning
Maria Termini
Brooklyn Law School
, maria.termini@brooklaw.edu
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52 Suffolk U. L. Rev. 5 (2019)
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Proving
the Point:
Connections
Between
Legal
and
Mathematical
Reasoning
Maria
Termini*
ABSTRACT
"I
think
there
are
a
lot
of
people
who
go
to
law school
because
they
are
not
good
at
math
and
can't
think
of
anything
else
to
do."'
Chief
Justice John
Roberts
got
a
good
laugh at that
line
when speaking
to
students
at
Rice
University. The
stereotype
about lawyers
disliking math
is
widely held,
including
by
lawyers
themselves.
But
the
disciplines
of
law
and
mathematics
are
more
closely
connected than
many
lawyers
suspect.
If
those
lawyers
who
are
"not
good
at
math"
took
an
upper-level
college
mathematics
course-one
focusing more
on
theorems
and
reasoning
than
on
numbers
and
calculations-they
might
find
it
far
easier
and
more
familiar
than
they
expect.
This
Article
explores
the
connections
between
mathematical
analysis and
legal
analysis-including
their
similar
organizational
schemes,
types
of
reasoning,
and
purposes-and
draws
lessons
for
legal
writing
from
the
realm
of
mathematics.
I.
INTRODUCTION
Fermat's
Last
Theorem
was
a
famous
unsolved
problem
for three
and
a
half
centuries,
and
its
solution made
headlines
even
outside
of
mathematical
circles.
The
Theorem takes
its
name
from
Pierre
de
Fermat,
an
avid amateur
mathematician
who
died
in
1665.2
After
Fermat's
death,
among
his
possessions
was
a
mathematics
text
that included
his
notes
related
to
the
Pythagorean
Theorem. The
Pythagorean
Theorem
gives
a
formula
for
determining
the
lengths
of
the
sides
of
a
right
triangle.
If
c
is
the
hypotenuse
of
a
right
triangle-i.e.,
the
side
opposite
the
right
angle-and
a and
b
are
the other
two sides
of
the
right
triangle,
then a
2
+
b
2
=
C
2
.
3
While
the
lengths
of
the
sides
of
a
right
triangle
are
*
Assistant Professor
of
Legal
Writing,
Brooklyn Law
School.
I
am
grateful
to
David
J.
Ziff,
K.
Sabeel
Rahman,
Jocelyn
Simonson,
Jayne
Ressler, Heidi
K.
Brown,
Meg
Holzer,
Natalie
Chin,
Kate Mogelescu,
Carrie
Teitcher,
Joy
Kanwar, and Rebecca
Rogers for their helpful comments,
to
Kathleen Darvil, Sara-Catherine
Gerdes,
and
Lauren
Cedeno
for
their
research
assistance,
and
to
the
Brooklyn
Law School
Dean's
Summer
Research Stipend Program
for financial
support.
1.
Rice
Univ.,
Centennial
Lecture Series:
ChiefJustice
John
Roberts
Speaks
at
Rice
University,
YouTube
(Oct.
26,
2012),
https://www.youtube.com/watch?v=UxaFhJ8JVq8
[https://perma.cc/4CAJ-UXQJ].
2.
SIMON SINGH,
FERMAT'S
ENIGMA:
THE
EPIC
QUEST
TO
SOLVE
THE
WORLD'S
GREATEST
MATHEMATICAL
PROBLEM
35,
62
(1997).
3.
See
id.
at
18-19
(explaining
Pythagorean Theorem).
SUFFOLK
UNIVERSITYLA
WREVIEW
not
always
whole numbers,
it
is
not
difficult
to
find
whole
numbers
for
a,
b,
and
c
that
fit
the
formula.
For
example,
a,
b,
and
c
could
be
3,
4,
and
5,
respectively,
because
32
+
42
=
52.
Another
example
is
5,
12,
and
13.
These
are
known
as
Pythagorean
triples.
4
The
Pythagorean
Theorem
and
Pythagorean
triples
appear in
Arithmetica,
a
book
written
in
approximately
250
C.E.
by
Diophantus,
a
scholar
in
Alexandria.
5
Centuries
later,
while
studying
his
Latin
copy
of
Arithmetica,
Fermat
considered
the
Pythagorean
Theorem
and variations
of
Pythagorean
triples.
6
Specifically,
he
wondered
whether
any
solutions
could
be
found
for
a,
b,
and
c
when
they
are
raised
to
any
power
greater
than
two.7
Are
there any
whole
number
solutions
when
a,
b,
and
c
are
each
cubed?
When
they
are
each
raised
to
the
fourth
power
or
the
fifth
power?
Fermat thought
the
answer
to
each
question
was
"no."'
In
the
margin
of
his
copy
of
Arithmetica,
Fermat
stated
that
for
any
whole
number
n
greater
than
two,
there
are
no
whole
numbers
a,
b,
and
c
that
make
the
equation
an
+
bn
=
c'
true.
9
Then,
in
Latin,
Fermat
noted:
"I
have
a
truly
marvelous
demonstration
of
this
proposition
which
this
margin
is
too
narrow
to
contain."'
0
After
Fermat's
death, his
son
found
and
published
this
and
other
mathematical
notes,
but
Fermat's
proof
of
his
"last
theorem"
was
never
found."
For
decades,
and
then
centuries,
other
mathematicians
tried
to
recreate
or
discover
the
proof
for
themselves,
without
success.1
2
In
1986,
Andrew
Wiles,
then
a
professor
of
mathematics
at
Princeton
University,
set
himself
the
task
of
proving
Fermat's
Last
Theorem.'
3
He
worked
on
the
project
for
seven
years,
mostly
alone
and
in
secret.1
4
In
1993,
Wiles
4.
See
id.
at
28
(defining
and providing
examples
of
Pythagorean
triples).
5.
See
id.
at
51,
60
(describing
life
and
work
of
Diophantus
of
Alexandria).
6.
See
SINGH,
supra
note
2,
at
60-62
(describing
formulas Fermat
considered).
7.
See id.
at
61
(outlining
Fermat's
thought
process).
8.
See
id.
(quoting
and translating
Fermat's
assertion
no
such
whole
number solutions
exist
when
power
exceeds
two).
9.
See
id. (stating
Fermat's
theory).
10.
See
SINGH,
supra
note
2,
at
62
(quoting
and translating
Fermat's
marginal
notation).
11.
See
id.
at
62-63
(describing
search
for
proof
of
theorem).
12.
See
James
Gleick,
Fermat's
Theorem
Solved?
Not
This
Time,
N.Y.
TIMES
(Mar.
29,
1988),
https://www
.nytimes.com/1988/03/29/science/fermat-s-theorem-solved-not-this-time.html
[https://perma.cc/GU5Q-UT6T]
(noting
"[a]mateurs
and
professionals
alike
...
struggled for
350
years"
attempting
to
prove
Fermat's
Last
Theorem).
13.
See
SINGH,
supra
note
2,
at
204-05
(noting
Wiles
inspired
by
another
mathematician's
proof
linking
Fermat's
Last
Theorem
to
Taniyama-Shimura
conjecture).
14.
See
Gina
Kolata,
How
a
Gap
in
the
Fermat
Proof
Was
Bridged,
N.Y.
TIMES
(Jan.
31,
1995),
https://www.nytimes.com/1995/01/31/science/how-a-gap-in-the-fermat-proof-was-bridged.html
[https://perma.
cc/57U7-W95T].
The story
of
Wiles
working
alone
fits
the
popular
image
of
a
mathematician
in
solitude,
but
that
image
is
largely
a
myth.
See
generally
REUBEN
HERSH
&
VERA JOHN-STEINER,
LOVING
AND HATING
MATHEMATICS:
CHALLENGING
THE
MYTHS
OF
MATHEMATICAL
LIFE
chs.
5,
6
(2011) (refuting
myth
of
mathematicians
working
in
isolation).
A
mathematician
at the
opposite
end
of
the
teamwork
spectrum
was
Paul
Erd6s,
who
was
known
to
be
a
prolific
collaborator.
See
id.
at
191.
Much
like
six
degrees
of
separation,
mathematicians
refer
to
their
Erdbs
numbers
to
describe
how
closely connected
they
are
to
Erd6s
through
[Vol.
LII:5
6
2019]
CONNECTIONSBETWEENLEGAL
AND
MATHEMATICAL
REASONING
announced
a
solution
in
a
series
of
three
lectures
at
Cambridge
University,
but
the
proof
had
to
be reviewed
by
referees before
publication.'
5
As
with
everything
related
to
Fermat's
Last
Theorem,
the
peer
review
process
was
not
easy.
During
the
review
process,
one
of
the
referees
discovered
a
problem
with
the
proof.16
Wiles
went
back
to
work
to
try
to
address
the
gap the
referee
had
identified.'
7
With
another
year
of
work,
and
in
collaboration
with his
former
student,
Richard
Taylor,
Wiles
was
able
to
bridge
the
gap,
though
he
nearly
gave
up
in
the
process.'
8
Ultimately,
the
proof
was
published
as
two separate
but
related
papers,
which
together
totaled
130
pages
and
filled
the
May
1995
volume
of
the
Annals
of
Mathematics.'
9
Andrew
Wiles
is
a
professional
mathematician.
His
entire
career
has
consisted
of
studying
and
teaching
mathematics
in
academia.
For
Pierre
de
Fermat,
on
the
other
hand,
mathematics
was
a
hobby,
though
a
very
serious
one;
Fermat's
career
was
in
the
law.
2
0
This
connection
between
law
and
mathematics
in
Fermat's
life
is
not
as
surprising
as
it
might
seem
at
first blush.
Legal
reasoning
and
mathematical
reasoning
are
profoundly
connected.
Abraham
Lincoln
explained
that
studying
mathematics
helped
him
better
prepare
for
his
legal
studies:
In
the
course
of
my
law-reading
I
constantly
came
upon
the
word
demonstrate.
I
thought,
at
first,
that
I
understood
its
meaning,
but
soon
became
satisfied
that
I
did
not.
...
At
last
I
said,
'L[incoln],
you
can
never
make
a
lawyer
if
you
do
not
understand
what
demonstrate
means;'
and
I
left
my
situation
in
Springfield,
went
home
to
my
father's
house,
and
staid
there
till
I
could
give
any
propositions
in
the
six
books
of
Euclid
21
at
sight.
I
then
found
out
what
'demonstrate'
means,
and
went
back
to
my
law
studies.
22
scholarly
collaborations.
Id.
at
155.
Someone
who
collaborated
with
Erdbs
has an
Erdbs
number
of
1,
someone
who
collaborated
with one
of
Erdbs's
collaborators
has
an
Erd6s
number
of
2,
and
so on.
Id.
15.
See
SINGH,
supra
note
2,
at
244,
253
(describing
announcement
and
publication
process).
16.
See
id
at
257
(relating
referee's
communications
with
Wiles
about
error).
17.
See
id.
at
259
(describing
Wiles's
initial
attempts
to
fix
error).
18.
See
Kolata,
supra
note
14
(detailing
Wiles's
successful
efforts
to
mend
his
proof).
19.
See
SINGH,
supra
note
2,
at
276-79
(explaining
publication
structure
of
Wiles's
proofs).
See
generally
Richard
Taylor
&
Andrew
Wiles,
Ring-Theoretic
Properties
of
Certain
Hecke
Algebras,
141
ANNALS
MATHEMATICS
553
(1995);
Andrew
John
Wiles,
Modular
Elliptic
Curves
and
Fermat's
Last
Theorem,
141
ANNALS
MATHEMATICS
443
(1995).
20.
SINGH,
supra
note
2,
at
35-36
(outlining
Fermat's
career).
21.
Euclid
was
the
Greek
mathematician
and
author
who
systematically
explained
mathematics
in
the
several
books
of
his
work,
Elements.
See
Jeremy
Gray,
Geometry,
in
THE
PRINCETON
COMPIANION
TO
MATHEMATICS
83,
83-84
(Timothy
Gowers
etal.
eds.,
2010).
Euclid
begins
Elements
with
some
basic
definitions
and
postulates,
which
are
statements
that
seem
to
be
self-evident
but
cannot
be
proven.
See
id.
at
84;
see
also
H.S.M.
COXETER,
NON-EUCLIDEAN
GEOMETRY
1-2
(6th
ed.
1998)
(explicating
nature
of
postulates).
Euclid
then
explains
geometry
by
starting
with
those
definitions
and
postulates
and
using
them
to
prove
theorems.
See
Gray,
supra,
at
84.
Thus,
Euclid's
Elements
consists
largely
of
mathematical
proofs.
Id.
22.
See
Mr.
Lincoln's
Early
Life.;
How
He
Educated
Himself
N.Y.
TIMES
(Sept.
4,
1864),
https://www.nyti
mes.com/1864/09/04/archives/mr-lincolns-early-life-how-he-educated-himself.html
[https://perma.cc/C9WK-JJ
7
SUFFOLK
LNIVERSITYIA
WREVIEW
Lincoln's
idea-to
study
mathematical reasoning
in
order
to
become
better
at
legal
reasoning-might
seem
strange,
especially
to
those lawyers
who
do
not
like
math.
But
law
and
mathematics
have
much
in
common, starting
with
the basic
organizational
structure
of
written analysis
in
each
field.
The logical
structure
of
written
proofs
in upper-level
mathematics
bears
a
close
resemblance
to
the
basic
organizational
structure
in
legal
writing:
Issue,
Rule,
Application,
Conclusion-often
referred
to
by
its
acronym
IRAC.
With a
proof,
as
in
IRAC,
a
writer
starts
by
telling
the
reader
where
the
writer
wants
to
go
(I);
then
lists
the
rules
or
theorems
already
known
(R);
applies
the
known rules
to
the
facts
or
"givens"
(A);
and
reaches
a
conclusion based
on that
reasoning
(C).
2 3
In
addition
to
using
similar
organization,
both
legal
and
mathematical
analysis
use
the
same
types
of
reasoning,
including
deductive
reasoning,
inductive
reasoning,
and
arguments
in
the
alternative.
Written
legal
analysis
also
has
similar
purposes
to
those
of
written mathematical
analysis.
Both
forms
of
writing
aid
the
author's
thought
process
and
are
used
to
convince
the
reader
that
the
author
is
correct, to extend the body
of
knowledge
within
the
field,
and
to
teach
those
new
to
the
field.
While
a
comparison
between
law and
mathematics
may seem
inapt
because
law
is
less
determinate
and
less
certain than
mathematics,
a
closer
look
at
mathematics
reveals this
lack
of
certainty
is
yet
another
point
of
similarity
between
these
two
fields.
24
Mathematics,
far
from
revealing
some
universal
truths,
can
only
establish
truths
based
on
the
assumptions
made
within
the
mathematical
system
being
used.
Similarly,
lawyers cannot
find
"the
law"
by
looking
for
universal truths,
but
instead must
consider
the
laws
as
they
exist
within
a
given
legal system.
25
Furthermore,
just
as
existing
enacted
law
and
common
law
cannot address
every
possible
scenario
that
may
come
before
a
court, it is impossible
to
develop
a
robust
mathematical system
in
which
all
mathematical
statements
can
be
proven
to
be true
or
false.
This
Article
proceeds
as
follows.
Part
II
explores
the
similarities
between
mathematical
analysis
and
legal
analysis, including
their
similarities
of
organization,
purpose,
and
types
of
reasoning. Part
III
addresses
potential
differences
between
law
and mathematics, concluding that those
differences
are
not
as
great
as
they
appear.
Part
IV
considers explanations
from
cognitive
science
about
the
effectiveness
of
these
forms
of
written
analysis.
Part
V
draws
lessons
for
legal
analysis
in
light
of
the
comparison
to
mathematical
analysis.
GC]
(recounting
J.P.
Gulliver's
conversation
with
Abraham Lincoln).
23.
See
infra
Section
H.A.2
(describing
IRAC
organization
structure
and
several
variations
on
basic IRAC
format).
24.
See
infra
Part III
(evaluating
similarities
between
law and mathematics).
25.
See
generally
H.
L.
A.
Hart, Positivism
and
the
Separation
ofLaw and
Morals,
71
HARV.
L.
REV.
593
(1958)
(discussing
ideals
of
legal
positivism).
8
[Vol.
LII:5
2019]
CONNECTIONS
BETWEEN
LEGAL
AND
MATHEMATICAL
REASONING
II.
How
Is
LEGAL
ANALYSIS
LIKE
MATHEMATICAL
ANALYSIS?
Written
legal
analysis
and
mathematical
proof
are
similar
in
several
ways.
In
addition
to
organizational
parallels,
they
also
use
similar
reasoning
and
share
many
comparable
purposes.
A.
Organization
1.
Mathematics
Most
high
school
students
in
the
United
States
learn
the
"two-column
proof'
in
geometry
class.
Two-column
proofs
provide
a
structure
in which
students
can
work
through
and explain
their
reasoning
when
proving
a
theorem.
A
two-
column
proof
typically
begins
with
the
"givens"--the
facts
that
should
be
assumed-and
a
statement
of
what
will
be
proven.
After
that,
the
proof
is
laid
out
in two
columns.
The
left-hand
column
contains
numbered
statements,
which
show
the
proof
writer's
reasoning,
starting
with
the
given
facts
and
building
on
those
facts
one
step
at
a
time
by
applying
rules
to
the
facts.
For
each
statement
in
the
left-hand
column,
the
right-hand
column
contains
the
"rule"
that
was
applied
to
reach
the
statement.
Each
rule
must
be
a
given,
a
definition,
a
property,26
a
postulate,
an
axiom,27
or
a
previously
proven
theorem.
28
The
proof
proceeds
step-by-step,
ending
when
the
fact
to
be
proven
is
the final
statement
in the
left-hand
column
and
it
is
justified
by
a
valid
reason
in
the
right-hand
column.
Figure
1,
below,
shows
an
example
of
a
simple
two-column
proof.
29
26.
The
properties
of
equality
are
basic
statements
that
are
always
true
of
equations.
For
example,
the
most
basic
property,
the
Reflexive
Property
of
Equality,
states
that,
for
any
number
x,
x
=
x.
In
other
words,
any
number
is
equal
to itself.
See
generally
Properties
ofEqualities,
MATH
PLANET,
https://www.mathplanet.com/ed
ucation/algebra-1/how-to-solve-linear-equations/properties-of-equalities
[https://perma.cc/53AM-GE7D].
27.
Postulates
and
axioms
are
statements
that
are
assumed
to
be
true
without
proof.
For
example, one
postulate
in
Euclidean
geometry
states
that
a
straight
line may
be
drawn
through
any
two
points.
See
CoXETER,
supra
note
21,
at
1
(identifying
Euclid's
postulates).
28.
See
RICHARD
HAMMACK,
BOOK
OF
PROOF
87
(2d
ed.
2013)
(discussing
key
terms
in
proofs).
"A
theorem
is
a
mathematical
statement
that
is
true
and can
be
(and
has
been) verified
as
true.
A
proof
of
a
theorem
is
a
written
verification
that
shows
that
the
theorem
is
definitely
and unequivocally
true."
Id.
The difference,
then, between
postulates
and
theorems
is
that
theorems
can
be
proven
true,
while
postulates
are accepted
as
true
but
cannot
be
proven.
See
supra
note
27
and
accompanying
text
(explaining
postulates).
29.
The
reasons
below
are
presented
in
a
way
typical
of
a
high
school geometry
class,
using
definitions
and
properties
that would
be
familiar
to
students
taking
the
class.
For
readers
who
may
have
imperfect
memories
of
their
high school
days,
this
Article
includes
additional
explanations
in
footnotes.
9
SUFFOLK
UNIVERSITYLA
WREVIEW
Figure
1:
Two-Column
Proof
Given:
Angle
A
and
Angle
B
are
supplementary
angles.
Angle
B
and
Angle
C
are
supplementary angles.
Prove:
The
measure
of
Angle
A
=
the
measure
of
Angle
C.
30
Statements
1.
Angle
A
and
Angle
B
are
supplementary
angles.
2.
Angle
B
and
Angle
C
are
supplementary
angles.
3.
The
measure
of
Angle
A
+
the
measure
of
Angle
B
=
180
degrees.
4.
The
measure
of
Angle
C
+
the
measure
of
Angle
B
=
180
degrees.
5.
The
measure
of
Angle
A
+
the
measure
of
Angle
B
=
the measure
of
Angle
C
+
the
measure
of
Angle
B.
6.
The
measure
of
Angle
A
=
the
measure
of
Angle
C.
Reasons
Given
Given
Supplementary
Angles
Axiom
31
Supplementary
Angles
Axiom
Transitive
Property
of
Equality
32
Subtraction
Property
of
Equality"
After
geometry,
many
students
never
see
mathematical
proofs
again.
The
proof, however,
is
an
important
part
of
advanced mathematics.
In
upper
level
mathematics
courses-and
with
mathematicians
generally-written
proofs
in
paragraph
form
are
preferred
over
two-column proofs.
In many
respects,
these
"paragraph"
proofs
are
like
two-column
proofs.
They
typically begin
with
the
givens
and
with
a
statement
of
what
is
to
be proven, and
move
on
to
a
series
of
statements and
the
justifications
for
those
statements,
building logically
to
the
conclusion.
3 4
The
main
distinction
is
the
form
of the
30.
This Article
uses
basic mathematical
symbols such
as
the
equal sign,
the plus
sign,
and
the
like.
However,
in
the interest
of
clarity
for
a
non-mathematical
audience,
this Article omits other
standard
mathematical
symbols
and instead
uses words and
phrases
to
convey
the same
meaning.
For
example,
this
Article
uses
the
phrase
"the
measure
of
Angle
A"
rather
than
the
symbolic representation
of
that
phrase.
31.
See
WALTER
J.
MEYER,
GEOMETRY
AND
ITS
APPLICATIONS
39
(2d
ed.
2006)
(stating
Supplementary
Angles
Axiom).
"If
two
angles
are
supplementary,
then their
measures
add
to
180[]
[degrees]."
Id.
32.
The
Transitive
Property
of
Equality
states
that,
for
any
numbers
x,
y,
and
z,
ifx
=
y
andy
=
z,
then x
=
z.
In
other
words,
if
two
numbers
are
equal
to
the
same
third
number,
then
they
are
also equal
to
each other. Here,
since
steps
three
and
four
each
contain
an
expression
that
is
equal
to
180
degrees,
those
two
expressions
are
also
equal to
each
other.
33. The
Subtraction Property
of
Equality
states that,
for
any
numbers x,
y,
and
z,
if
x
=
y,
then
x
-
z
=y
-
z.
In
other
words,
if
two
numbers
are
equal
and you
subtract
the
same
thing
from each
number,
the
results
will
also
be
equal. Here,
the
measure
of
Angle B is subtracted from each
side
of
the equation.
34.
See
EUGENIA
CHENG, UNIV.
OF
CHI.,
DEP'T
OF
MATHEMATICS,
How
To
WRITE
PROOFS:
A
QUICK
GUIDE
3
(2004),
http://cheng.staff.shef.ac.uk/proofguide/proofguide.pdf
[https://perma.cc/6JYU-4B26]
(explaining steps
to writing
proofs).
"A
proof
is
a
series
of
statements, each
of
which follows
logically
from
what
has
gone
before. It
starts with
things
we
are
assuming
to
be
true.
It ends with
the
thing
we
are trying
to
10
[Vol.
LII:5
2019] CONNECTIONSBETWEENLEGAL
AND
MATIEMATICAL
REASONIVG
statements
and
justifications
of
each
step
of
the
proof.
Instead
of
listing
those
steps
rigidly
in
a
two-column
format,
mathematicians
usually
explain
themselves
using
complete
sentences
and
paragraphs.
35
Figure
2,
below, shows
an
example
of
a
"paragraph"
proof.
Figure
2:
Proof
Theorem:
If
x
and y
are
even integers,"
6
then
the
sum
of
x
and
y
is
even.
3
Proof
Suppose
x
and y
are
even
integers.
38
Since
x
is
an
even
integer, there
must
be
an
integer
n
such
that
x
=
2n."
Similarly,
since
y
is
also
even,
there
must
be
an
integer
m
such
that
y
=
2m.
Thus,
x
+
y
=
2n
+
2m
=
2(n
+
m).
Since
n
and
m
are
both
integers,
n
+
m
must
also be an
integer.
Then,
since
x
+
y
=
2(n
+
m)
and n
+
m
is
an integer,
x
+
y
is
even.'
2.
Law
This
basic
proof
structure-starting
with
the
thing
to
be
proven,
stating
rules
and
applying
them
to
the
facts,
and then
reaching
a
conclusion-bears
a
remarkable
resemblance
to
the IRAC
organizational
scheme
used
in
legal
writing.
41
Just
as
most
geometry
students
will
learn
about
two-column
proofs,
most
beginning
law
students
will
learn about IRAC.
IRAC
is
an
acronym that
prove."
Id.
In
some
proofs,
mathematicians
prove
supporting
theorems
along
the
way
to proving
the
main
theorem
that
is
the
goal
of
the
proof;
these
supporting
theorems
are
called lemmas.
See
HAMMACK,
supra
note
28,
at
88.
"A
lemma
is
a
theorem whose
main
purpose
is
to
help
prove
another
theorem."
Id.
35.
See
HAMMACK,
supra
note
28,
at
94
(noting
process
of
generating
and writing
proof
ends
by
"writ[ing]
...
proof
in
paragraph
form").
"One
doesn't
normally
use
a
separate
line
for
each
sentence
in
a
proof
....
"
Id.
at
96.
36.
Integers
are
positive
and
negative
whole
numbers.
See
id
at 4.
37.
This
formulation-a
statement
of
the
theorem
being
proven-combines
the
givens
with
the
thing
to
be
proven.
In
the
standard two-column
proof
format, the
theorem statement
could
have
been
rewritten
as
follows:
Given: x
and
y
are
even
integers.
Prove:
the
sum
of
x
and
y
is
even.
38.
The
proof
begins
by
assuming that the
givens
are
true.
39.
This
sentence follows
from
the definition
of
an
even
number.
If
asked
to
define "even
number,"
most
people
would
likely
say
that
an
even
number
is
a
number
that
is
divisible
by
2.
In
proofs, mathematicians
use
a
somewhat more
formal
version
of
that
definition: "An
integer
n
is
even
if
n
=
2a
for
some
integer
a."
See
HAMMACK,
supra
note
28,
at
89.
This
definition
can
be
useful
in
proofs
because
it
allows
the
writer
to
consider
and
perform operations
on
both
n
and
a.
Unlike
in
a two-column
proof,
where
each
statement
must
have
a
justification,
here, some
justifications
are
sufficiently
evident-to
a
knowledgeable
mathematical
reader-to
be
omitted.
See
id.
at
87.
"A
proof
should
be
understandable
and
convincing
to
anyone
who
has
the
requisite
background
and
knowledge. This
knowledge
includes
an
understanding
of
the
meanings
of
the
mathematical
words,
phrases
and
symbols that
occur
in
the
theorem
and
its
proof."
Id.
40.
The
proof
has
established
that
the
sum
of
x
and
y
is
even.
It
has
therefore
reached
the
intended
conclusion
and
is
complete.
41.
See
infra
notes
47-54
and
accompanying
text
(discussing IRAC
and
its
variations).
11
2019]
CONNECTIONS
BETWEEN
LEGAL
AND
MATHEMATICAL
REASONING
formed
a
common-law
marriage
while in
Rhode
Island,"
will
focus
the
reader
on
the
writer's
conclusion
on
the
same
issue.
Beyond
the
switch
to
CRAC
for persuasive
writing,
many
IRAC
variations
attempt
to
expand
the structure
in
various
ways
to
help
the
students
fully
flesh
out complex
legal
arguments.
Consider
the
IRAC
example above
in
Figure
3:
The
facts
to
which
the
rules
are
applied
are
very
one-sided;
the
facts
are
strongly
supportive
of
a
common-law
marriage
and
there
are
no
facts
that
would
indicate
a
common-law
marriage
had
not
been
established.
With
other, more
equivocal
facts,
a
short
IRAC
would
be
insufficient.
Consider,
for example,
if
instead
of
promising
to
be
"husband
and
wife,"
the
couple
had
promised
to
be
"together
for
life." Would
that
be
enough
to
show
the
requisite
"mutual
present
intent"
to
enter
a
spousal
relationship?
Maybe,
maybe
not,
but
the
reader
will
probably
need
more
information.
The
writer
should
look
at
case law
to
see
whether
courts
have
given
more
information
about
what
shows
"mutual
present
intent."
If
two
of
the
witnesses
at
trial
testified
that
they
considered
the
couple
to
be dating
rather than
married,
would
there
still
be enough
proof
of
a
community
belief
that
they
were
spouses?
Perhaps.
The
writer
should
again
look
at
the
precedent
to
see
whether previous
cases can
help
answer
that
question.
In
order
to
fully
analyze
the
scenario
with more
equivocal
facts,
the
writer
will
need
to
expand
the
R
and
A
sections
of
IRAC.
Over
the
years,
scholars have
created
new
acronyms
to
help show
students
how best
to
use
IRAC
when
the
reader
will
need
more detail and
explanation.
These
variations
on
IRAC
include
CREAC
(Conclusion,
Rule,
Explanation,
Application,
Conclusion);
48
CRuPAC
(Conclusion,
Rule,
Proof
of
Rule,
Application,
Conclusion);
49
TREAT
(Thesis,
Rule,
Explanation,
Application,
Thesis);
5
0
CREXAC
(Conclusion,
Rule,
Explanation,
Application,
Conclusion);s'
and
CRRPAP
(Conclusion,
Rule,
Rule
Proof,
Application,
Prediction).
52
Many
of
these
IRAC
variants
provide
a
place
for
a
deeper
exploration
of
the
rule
at
issue.
For
example,
in
CREAC,
the
added
E
is
for
"explanation"
of
the
rule.
3
For
students
new
to
legal
writing,
this
often
centers
48.
See
HEIDI
K
BROWN,
THE
MINDFUL
LEGAL
WRITER
MASTERING
PREDICTIVE
WRITING
91
n.1
(2015)
(discussing
CREAC
as
alternative
to
IRAC).
49.
See
RICHARD
K.
NEUMANN,
JR.,
LEGAL
REASONING
AND
LEGAL
WRITING:
STRUCTURE,
STRATEGY,
AND
STYLE
94
(6th
ed.
2009)
(detailing
elements
of
CRuPAC).
50.
See
MICHAEL
D.
MURRAY
&
CHRISTY
HALLAM DESANCTIS, LEGAL
WRITING
AND
ANALYSIS
139
(2009)
(identifying elements
of
TREAT).
51.
See
MARY BETH
BEAZLEY,
A
PRACTICAL
GUIDE
TO
APPELLATE
ADVOCACY
77
(3d
ed.
2010)
(listing
elements
of
CREXAC).
52.
See
CHRISTINE
COUGHLIN
ET
AL.,
A
LAWYER
WRITES:
A
PRACTICAL
GUIDE
TO
LEGAL
ANALYSIS
83
(2d
ed.
2013)
(defining
CRRPAP).
Further
variations
include
BaRAC,
CRARC,
IGPAC,
IRAAC(P),
IRAAPC,
IRAC
with
ElP,
IRAC
plus,
IRRAC,
RAFADC,
TREAC,
and
TRRAC.
See
Tracy
Turner,
Finding
Consensus
in
Legal
Writing
Discourse Regarding
Organizational
Structure:
A
Review
and
Analysis
of
the
Use
ofIRAC
and
Its
Progenies,
9
LEGAL
COMM.
&
RHETORIC:
JALWD
351,
357-58
(2012)
(displaying
table
of
IRAC
variant
acronyms).
53.
See
BROWN,
supra
note
48,
at
91
n.1,
94.
13
SUFFOLK
UNIVERSITY
LAW
REVIEW
on
the
legal
precedent
that
will
later be
used
in
A.
In
E,
for
each
precedent
case,
the
writer
gives
the
facts,
reasoning,
and
holding
that
are
relevant
to
the
issue
the
IRAC
addresses.
54
Fleshing
out
the
cases
in
this
way
helps
the
reader
see
how
courts
have
interpreted
and
applied
the
rules
in
R,
and
allows
the
writer
to
later
analogize
or
distinguish
the
case
at
hand
and
the
precedent
cases
in
A.ss
Since
the
information
in
E
is
detailed
information
about
the
rule,
it
could be
considered
part
of
R.
56
Many
legal
writing
professors,
however,
find
it
useful
to
include
E
in
the
acronym
so
that
students
remember
to
include
this
information.
The
information
in
E
is
not
limited
to
the
facts,
reasoning,
and
holding
of
precedent.
As
appropriate,
E
can
be
used
for
other
details
that
help
explain
the
rule.
For
a
statutory
rule,
for
example,
E
might
include
legislative
history
relevant
to
the
issue
at
hand.
B.
Types
ofReasoning
As
noted
earlier,
lawyers
and
mathematicians
use
similar
reasoning
in
their
thought
processes
and
writing.
In
considering
problems,
both
mathematicians
and
lawyers
use
deductive
reasoning,
inductive
reasoning,
and
arguments
in
the
alternative.
1.
Deduction
At
their
cores,
both
mathematical
proofs
and
IRAC
use deductive
reasoning
and
syllogisms.
57
The
syllogism
is
a
structured
form
of
deductive
reasoning-
reasoning
from
the
general
to
the
specific-involving
three
parts:
the
major
premise,
which
is
the
general
rule;
the
minor
premise,
which
consists
of
the
specific
facts
at
hand;
and
the
conclusion
about
the
specific
facts
in
light
of
the
general
rule.
5
8
Figure
4,
below, shows
a
well-known
syllogism.
59
Figure
4:
Syllozism
Example
Step
1.
All
humans
are
mortal.'
Major
premise
Step
2.
Socrates
is
human.
Minor
premise
Step
3.
Therefore,
Socrates
is
mortal.
Conclusion
54.
See
id
at
112
(addressing
use
of
precedent
in
explanation
section).
55.
See
id
at
146
(discussing
ways
writer
can
compare
and
contrast
precedent).
56.
The information
in
E
helps
establish
the
"lemmas"
of
legal
writing-the
"theorems"
that
help
prove
the
main
"theorem."
See
supra
note
34
and
accompanying
text (defining
"lemmas").
57.
See
KRISTEN
KONRAD
TISCIoNE,
RHETORIC
FOR
LEGAL
WRITERS:
THE
THEORY
AND
PRACTICE
OF
ANALYSIS
AND
PERSUASION
88
(2d
ed.
2016)
(explaining
IRAC
is
"shorthand
for
deductive
reasoning").
58.
See
id. at
87-88
(explaining
syllogisms and
deductive reasoning).
59.
See
id.
at
88.
60.
This
statement
can
be
reworded
to
fit
the
if-then
format,
also
known
as
a
conditional
statement:
If
a
thing
is
human,
then
that
thing
is
mortal.
See
infra
note
172
and accompanying
text
(explaining
conditional
statements).
[Vol.
LII:5
14
2019]
CONNECTIONS
BETWEEN
LEGAL
AND
MATHEMATICAL
REASONING
In
mathematics,
each
step
of
the
proof
should
rely on
a
syllogism.
For
example,
moving
from
Steps
3
and
4
to
Step
5
in
the
two-column
proof
above
61
involved
the
following
syllogism:
Figure
5:
Svlo2ism in
Mathematical
Proof
The
Transitive
Property
of
Equality
(for
any
numbers
x,
y,
Major premise
and
z,
ifx
=
y
andy
=
z,
then
x
=
z).
The
measure
of
Angle
A
+
the
measure
of
Angle
B
=
180
Minor
premise
degrees
and
the
measure
of
Angle
C
+ the
measure
of
Angle
B
=
180
degrees.
Therefore,
the
measure
of
Angle
A
+
the
measure
of
Angle
Conclusion
B
=
the
measure
of
Angle
C
+
the measure
of
Angle
B.
Similarly,
the
organization
of
IRAC
relies
on the
structure
of
syllogisms.
62
The
RAC
of
IRAC
is
a
syllogism.
6
3
Figure
6: Svllogism
in
IRAC
A
common-law
marriage
is
formed
when
(1)
a
couple
had
a
mutual
present
intent
to
enter
a
spousal
relationship
and
(2)
the
actions
of
the
couple
led
to
a
belief
in
the
community
that
they
were
in
a
spousal
relationship.
"
Smith
v.
Smith,
966
A.2d
109, 114,
117
(R.I.
2009).
Here,
Ms.
Garcia
and
Mr.
Jordan
had
a
mutual
present
intent
to
enter
a
spousal relationship. Furthermore,
their
actions
led
to
a
belief
in
the
community
that they
were
in
a
spousal
relationship.
Therefore,
Ms.
Garcia
and
Mr.
Jordan
did
form
a
common-
law
marriage.
Major
premise
Minor
premise
Conclusion
61.
See
supra
Figure
1.
62.
See
Anita
Schnee,
Logical Reasoning
"Obviously",
3
LEGAL
WRITING:
J.
LEGAL
WRITING
INST.
105,
106 (1997)
(noting
"'IRAC'
.
..
just
another way
of
saying...
'deductive syllogistic
process').
63.
See
Nelson
P.
Miller
&
Bradley
J.
Charles,
Meeting
the
Carnegie
Report's
Challenge
to
Make
Legal
Analysis
Explicit-Subsidiary
Skills
to
the
IRAC
Framework,
59
J. LEGAL
EDUC.
192,
208 (2009)
(explaining
IRAC
"mirrors
.
..
syllogism");
Turner,
supra
note
52,
at
356 (explaining
"adaptation
of
deductive
syllogism
to
legal
reasoning");
Penny
L.
Willrich,
The
Path
to
Resilience:
Integrating
Critical
Thinking Skills
into
the
Family
Law
Curriculum,
3
PHX.
L.
REv.
435,
451
(2010).
"[T]he
major
premise
is
based upon
the applicable
rules.
The
minor
premise
is
found
in
the
facts
or
evidence."
Willrich,
supra,
at
451.
64.
This rule
can
also
be
reworded
as
an
if-then
conditional statement:
If
a
couple
had a
mutual
present
intent
to
enter
a
spousal
relationship
and the
actions
of
the
couple
led
to a
belief
in
the
community that
they
were
in
a spousal
relationship,
then
the couple formed
a
common-law
marriage.
See
infra
note
172
and
accompanying
text
(detailing
conditional
statements).
15
SUFFOLK
UNIVERSITY
LAW
REVIEW
2.
Induction
While
deductive
reasoning
is
reasoning
from
the
general
to
the
specific-
applying
a
general
rule
to specific
facts-inductive
reasoning
is
reasoning
from
the specific
to
the
general-using
specific
facts
to
try
to
find
a
general
rule.
6
5
Mathematicians
use
induction
in
two
different,
but
related, ways.
George
Polya,
an
influential
mathematician
and
mathematics
educator
who
wrote
the
book
How
to
Solve
It:
A
New
Aspect
of
Mathematical
Method-which
has
helped
generations
of
mathematics
students
understand
mathematical
reasoning
and
proof-used
different
terms
for
the
two
types
of
induction.
66
He
distinguished
between
induction,
"the process
of
discovering
general
laws
by
the
observation
and
combination
of
particular
instances,"
and
mathematical
induction,
the
rigorous
method
of
proof
by
induction.
67
In
mathematical
induction,
i.e.,
a
proof
by
induction,
a
mathematician
uses
the
fact
that
a
statement
is
true
for
one
case to
show
that
it
must
be
true
for
all
cases.
68
Proof
by
induction
requires
two
steps.
69
First,
in
the
"basis
step,"
the
writer
shows
that
the
statement
is
true
for
one specific
case,
often
the
number
"1."70
Second,
in
the
"inductive
step,"
the
writer
first
assumes
that
the
statement
is
true
for
any
positive
integer
x
(the
"induction
hypothesis"),
and
then
shows,
based
on
that
assumption,
that
the statement
must
also
be true
for
the
integer x
+
171
If
the
writer
can
establish
both
of
those
things,
then
it
follows
that
the statement
is
true
for
all
positive
integers
because
it
is
true
for
"1";
the
fact it
is
true
for
"1"
implies it
is true
for
"2"
(based
on the
inductive
step);
the
fact it
is
true
for
"2"
implies
it
is
true
for
"3"
(again based
on
the inductive
step);
and
so
on.
A
common
analogy
is
to
toppling dominoes.
In
the
basis
step,
the writer
shows
that
the
first
domino
will
topple.
In
the
inductive
step,
the
writer
shows
that toppling
one
domino
will topple
the
next
domino.
Putting
those
two
facts
together
shows
that
all
the
dominos
will
topple.
Here
is
an
example
of
a
proof
by
induction:
65.
See
Schnee,
supra
note
62,
at
111
(comparing
deductive reasoning
and
inductive
reasoning).
66.
GEORGE
POLYA,
How
TO
SOLVE
IT:
A
NEW
ASPECT
OF
MATHEMATICAL
METHOD
114
(Anchor
Books
2d
ed.
1985).
67.
Id.
(describing types
of
induction).
68.
See
HAMMACK,
supra
note
28,
at
155
(giving
example
of
proof
by induction).
69. See
id.
at
156
(outlining
proof
by induction).
70. See
id.
(detailing
basis
step
of
proof
by
induction).
71.
See
id.
(explaining
inductive
step
and
induction hypothesis).
16
[Vol.
LH:5
2019]
CONECTIONS
BETWEENLEGAL
AND
MATHEMAICAL
REASONIVG
Figure
7:
Proof
by
Induction
Theorem:
For
any
positive
integer
n,
the
sum
of
the
integers from
1
through
n
equals
n(n
+
1)/2.
In
other
words,
1
+
2
+
3
+
n
=
n(n
+
1)/2.
Proof
[Basis
Step]
72
[Inductive
Step]
If
n
=
1,
then
the
sum
of
the
integers
1
through
n
is
1." When
n
=
1,
n(n
+
1)12
=
1(1
+
1)/2
=
1(2)/2
=
2/2
=
1.74
Therefore,
the
statement
1
+
2
. ..
+
n
=
n(n
+
1)/2
is
true where
n
=
1.75
Suppose
k
is
a
positive
integer
and that
1
+
2
+
3
+ . . . +
k
=
k(k
+
1)12.76
Then
this
proof
must
show
that
1
+
2
+
3
+ . . .
+
(k
+
1)
=
(k
+
1)((k
+
1)
+
1)12 =
(k
+
1)(k +
2)/2.
k(k
+
1)12
+
(k
+
1)
=77
k(k
+
1)/2
+
2(k
+
1)12
=
(k
+
2)(k
+
1)/2
7
Therefore,
based on
induction,
for
any
positive integer
n,
the
sum
of
the
integers
from
1
through
n
equals
n(n
+
1)12.
Mathematicians
also
use
inductive
reasoning
before the
stage
of
setting
down
a
formal
proof.
While
the
inductive
proof
above
was
proven
using mathematical
induction,
a
mathematician
could have
used
inductive
reasoning
to
generate
a
hypothesis
to
prove
in
the
first
place.
The thought
process might
go
like
this:
72.
For
the
basis
step,
the
writer
needs
to
show
that
equation
in
the
theorem
(1
+
2
+
3
+...
+
n
=
n(n
+
1)/2)
is
true
when
n
=
1.
73.
This takes
care
of
the
left
side
of
the
equation.
74.
And
this
solves
the
right
side
of
the
equation.
75.
This follows from
the
fact
that
the
previous
two steps
showed
that
both sides
of
the equation
equal
one
when
n
=
1.
76.
This
is
the
induction
hypothesis.
The
goal
now
is
to
prove
that
the statement holds
true for
(c
+
1),
i.e.,
1
+
2
+
3
+.
.. +
(k
+
1)
=
(k
+
1)((k
+
1)
+
1)/2.
77.
This
step
uses the assumption
in
the
induction hypothesis
to
replace
I
+
2
+
3
+ . .. +
k
with
the
expression we assumed
it
was
equal
to
in the
induction
hypothesis,
k(k
+
1)/2.
78.
This
step
multiplies
the
expression
(k
+
1)
by
2/2
-otherwise
known
as
1,
meaning
that the
value does
not
change-so
that
the
two
fractions
have
a
common
denominator.
79.
This step adds
the
fractions.
Having
reduced
the
expressions,
this
series
of
statements
has
shown
that
1
+
2
+
3
+.
. . +
(k
+
1)
=
k+)(+
k
+
2)/2,
which was
the
goal
of
the
inductive
step.
17
SUFFOLK
UNIVERSITYLA
WREVIEW
o
Figure
8:
Inductive
Reasoning
(the
thought
process
of
a
mathematician)
Is
there
a
general
formula
for
the sum
of
the
first
n
integers?
Well,
when
n
is
1,
the
sum
of
the
first
n
integers
is
just
1.
When
n
is
2,
the
sum
of
the
first
2
integers
is
1
+
2
or
3.
Let
me
look
at the
first
several
integers.
n
Sum
of
the
first
n
integers
1
1
2
1+2=3
3
1+2+3=
6
4
1+2+3+4=10
5
1+2+3+4+5=15
6
1
+2+3+4+5+6=21
7
1+2+3+4+5+6+7=28
Well
the
odd
numbers
are
interesting.
The
sums
are
multiples
of
n
when
n is odd,
at
least
for
these
numbers.
Let's
look
at
those
multiples.
n
Sum
of
the
first
n
integers
n
times
what?
1
1
1
3
1+2+3=6
2
5
1+2+3+4+5=15
3
7
1+2+3+4+5+6+7=28
4
Aha!
The
multipliers
follow
a
pattern!
In
fact,
the even
numbers
fit
the
pattern,
too,
it's
just
that
the
multipliers
are
not
whole
numbers.
n
Sum
of
the
first
n
intergers
n
times
what?
1
1
1
2
1
+
2
=
3
1.5
3
1+2+3=6
2
4
1+2+3+4=10
2.5
5
1+2+3+4+5=15
3
6
1+2+3+4+5+6=21
3.5
7
1+2+3+4+5+6+7=28
4
So,
then,
can
this
be
translated
into
a
general
formula?
It
appears,
for
each
n,
the
multiplier
is
half
of
the
next
n
on the list.
So
maybe
it
is
true for
any
integer
n that
the
sum
of
the
integers
from
1
through
n
equals
n(n
+
1)12.
The
inductive
thought
process yields
a
possible
formula
for
the
sum
of
the
first
n
integers,
but
it
does
not
prove
that
the
formula
always
holds
true.
It
is
possible
that
the
pattern
will
break
down
when
n
equals
8,
or
1,008,
or
some
other
number.
As
shown
in
the
proof
in
Figure
7,
however,
mathematical
induction
can
be
used
to prove
that
the
formula
is
correct
and
applies not
just
to
the
first
seven
integers,
but
to
all
positive
integers.
18
[Vol.
LII:5
2019]
CONNECTIONSBETWEENLEGAL
AND
MATHEMATICAL
REASONING
Although
lawyers
do
not
create
domino-toppling
proofs
by
induction,
they
do
use
inductive
reasoning much
like
the
mathematical
thought
process
above.
Lawyers
and
mathematicians
alike
can
use
induction-as
distinct
from
mathematical
induction-to
try
to
identify
general
rules
based
on
specific factual
scenarios.
While
mathematicians
use
inductive
reasoning
based
on
specific
numbers,
lawyers
use
inductive
reasoning
on
the
facts
of
specific cases.
For
lawyers,
inductive
reasoning
works
hand-in-hand
with
the common law
and the use
of
precedent.
Lawyers consider
the
facts
of
earlier
cases
to
try
to
find
a
pattern that
might indicate
a
general
principle
of
law
that
could
be
relevant
to the case
at
hand.
80
For
example,
a
lawyer analyzing
a
common-law marriage
issue
81
might
come
upon
cases
with
the
following facts
relating
to
the
community
belief:
Case
One:
Five
witnesses
testified,
including
two
family
members
of
the
putative
spouses.
All
five
stated that
they
thought
the couple
was
married
even
though
the
couple
used
different
last
names. The
court held
that there
was
a
community
belief
that
the couple
was
married.
Case
Two:
Three
witnesses
testified.
One
of
the
witnesses,
the
putative
husband's
brother,
stated that his
family-including
the
putative husband-
treated
the
woman
as
a
girlfriend
rather
than
a
spouse.
The
brother
also
testified
that
his
great-aunt,
the
family's
unofficial
genealogist,
did not
include the
woman
in
the
family
tree
she
had
created
six
months
before.
The
other
two
witnesses,
both
friends
of
the
couple,
stated
that
they believed
the
couple
to
be married. The
court
held that there
was
no
community
belief
that
the couple was married.
Case
Three:
Ten
witnesses
testified.
All
ten
stated
that
they
believed
the
couple
to
be
married
and
knew
the
woman
by
the
man's
last
name.
The
court held that
there was
a
community
belief
that
the
couple
was
married.
Case
Four:
Four
witnesses
testified.
The
first
witness,
the
couple's
upstairs
neighbor
and
landlord,
stated that
she
thought
the
couple
was
married.
The other
three
witnesses,
friends
of
the
couple,
testified that they
thought
of
the couple
as
in
a
long-term relationship but
not
a
marriage. The
court
held
that there was
no
community
belief
that
the couple
was
married.
A
lawyer,
reading
these four
cases,
could
use
inductive reasoning
to
put
together
the
facts
of
the
cases and write
the
following
in
a
memo
or
brief:
Where the
testimony and
evidence
is
consistent
that
community members
thought
a
couple
to
be
married,
courts
have held
that
there
is a
belief
in
the
community that
a
couple
is
married.
In
contrast,
when witnesses
disagree
as
to
80.
See
TISCIONE,
supra
note
57,
at
80.
81.
Cf
supra
Figures
3,
6
(providing
IRAC
and
syllogism
in
IRAC
regarding
common-law
marriage
issue).
19
SUFFOLK
UNIVERSITYLAWREVIEW
whether
a couple
is
married,
courts
have
held
that
there
is
no
belief
in
the
community
that
the
couple
is
married.
The
lawyer
could
then
compare
the facts
of
these
four
cases
to
the
facts
of
Ms.
Garcia
and
Mr.
Jordan's
case
to
analyze
how
the
case
should
be resolved.
If,
in
the
Garcia/Jordan
case,
all
seventeen
witnesses
at
trial
testified
that
they
considered
Ms.
Garcia
and
Mr.
Jordan
to
be
husband
and
wife,
the
lawyer
could
conclude
that
a
court
would
likely
hold
that
there is
a
community
belief
in
their
spousal
relationship.
3.
Arguments
in
the
Alternative
One
variation
on
the
standard
mathematical
proof
is
a
proof
that
uses
"cases."
In
some
situations,
it
is
not
possible
to
write
a
proof
that
establishes
the
truth
of
a
statement
in
all
conditions.
Instead,
the
writer
of
the
proof
must
consider
two
(or
more)
cases
and
prove
the
statement
for
each
of
the
cases.
82
For
example,
the
cases
might
be
even
integers
and
odd
integers.
83
If
the
writer
can
show
that
a
certain
statement
is
true
for
all
even
integers
and
also
true
for
all
odd
integers,
then
the
writer
has
proven
the statement
to
be
true for
all
integers.
In
developing
a
proof
using
cases,
the
writer
must
be careful
in
selecting
the
cases
to
cover
all
the
possibilities.
Since
all
integers
must
be
either
odd or
even,
separately
proving
the
"odd
integers"
case
and
the
"even
integers"
case
will
validly
prove
that
the
statement
is
true
for
all
integers.
If,
on
the
other
hand,
the
writer
wanted
to
prove
that
a
statement
was true
for
all
real
numbers,
proving
that
it
was
true
for
all
odd
and
even
integers
would
not
be
sufficient
because
the
set
of
real
numbers
contains
non-integer
numbers
such
as
pi,
%,
and
-19.7.
Legal
writers
use
similar reasoning
to
argue
based
on
multiple
"cases,"
though
the
terminology
is
different.
Lawyers
argue
different
"cases"
when
they
make
arguments
in
the
alternative.
Consider
the
following
excerpt
from
a table
of
contents
of
a
federal
appellate
brief:
ARGUMENT
C.
Rodriguez
Has
Not
Met
His
Burden
of
Establishing
the
Existence
of
a
Plea
Agreement
D.
In
the
Alternative,
Even
If
the
Putative
Plea
Agreement
is
Assumed
to
Exist,
Arguendo,
Rodriguez
Has
Not
Met
His
Burden
of
Demonstrating
that
the
Government
Breached
That
Agreement
In
Any Way
82.
See
HAMMACK,
supra
note
28,
at
98.
"In
proving
a
statement is
true,
we
sometimes
have
to examine
multiple
cases
before
showing
the
statement
is
true
in
all
possible
scenarios."
Id.
83.
See
id.
[Vol.
LII:5
20
2019] CONNECTIONS
BETWEENLEGAL
AND
MATHEMATICAL
REASONING
G.
In
the Alternative,
Even
If
the
Putative
Plea
Agreement
is
Assumed
to
Exist,
Arguendo,
Rodriguez
Has
Not
Satisfied
the
Fourth
Prong
of
Plain
Error
Review8
Essentially,
the
argument
in
this
brief
considers
two
"cases":
there
is
no
plea
agreement,
and
there
is
a
plea
agreement.
In
section
C,
the
writer
argues
that
the court
should
affirm
the
lower
court
because
there
was no
plea
agreement
(the
first "case").
86
In
sections
D
through
G,
the
writer
argues
that
the
court should
affirm the
lower
court
even
if
there
was
a
plea
agreement
(the
second
"case").
8 7
4.
Reductio
Ad
Absurdum
Another
common
tool
in
the
mathematician's
toolkit
is
the
proof
by
contradiction.
This
type
of
proof
begins
with
an
assumption
that
the
conjecture
being proven
is
false.
88
Based
on
this
assumption,
the
writer
of
the
proof
moves
from
step
to
step
using
theorems, axioms,
and
deductive reasoning,
trying
to
reach
a
contradiction,
an
absurd
result.
89
If
the
steps
of
the
proof
do
lead to
a
contradiction, this
means
that
the
original
assumption-that
the
conjecture
was
false-must
itself
be
false,
and
therefore
the
conjecture
is
true.
0
Figure
9:
Proof
by
Contradiction
Theorem:
For
any integer n,
if
n'
is
odd
then
n
is
also
odd.
Proof
Suppose,
for
the
sake
of
contradiction,
that
n'
is
odd and
n
is
even.
If
n is
even,
then
there
exists
an
integer
k
such
that
n
=
2k.
9
'
Then:
n2=
n
*
n=
2k *
2k
=
4(k
*
k)=
2(2k *
k)=
(2k
*
k)
is an
integer
because it
is
the
product
of
integers.
Therefore,
based
on the
definition
of
even
numbers,
n
2
is
even since
n
*
n
=
2(2k
*
k).
This
is a
contradiction
since we
began
by
supposing
that
n
2
is
odd.
Therefore,
the
supposition
is
false
and
the
theorem must
be
true.
84.
Brief
for
Plaintiff-Appelee
at
iii,
United
States
v.
Rodriguez-Diaz, 694
F.
App'x
263 (5th
Cir.
2017)
(No.
16-41300),
2017
WL
1047797,
at
*iii.
85.
See
id. at
19-20,
23
(outlining
main arguments).
86.
See
id.
at
19-20
(arguing
defendant
failed to
meet
his
burden
of
establishing
existence
of
plea
agreement).
87.
See
id.
at
20-28
(making
alternative arguments).
88.
See
HAMMACK,
supra
note
28,
at
111
(describing
method
of
proof
by
contradiction).
89.
See
id.
(showing example
of
proof
by contradiction).
90.
See
id.
(describing
conclusions
that follow from
a contradiction).
91.
See
supra
note
38
and
accompanying text.
21
SUFFOLK
UNTVERSITYLA
WREVIEW
In
the
law,
writers
also
sometimes
reason
by
assuming
the
opposite
of
what
they
are
trying
to show.
For
example,
in
Corley
v.
United
States,
92
the
Supreme
Court
rejected
the
government's
argument
about
a
statutory
section
by,
among
other
things,
using
reductio
ad
absurdum.
93
The
Court
noted,
if
the
statute
was
read
literally,
as
urged
by
the
Government,
absurd
results
would
follow.
Thus
would
many
a
Rule
of
Evidence
be
overridden
in
case
after
case:
a
defendant's
self-incriminating
statement
to his
lawyer
would
be
admissible
despite
his insistence
on
attorney-client
privilege;
a
fourth-hand
hearsay
statement
the
defendant
allegedly
made
would
come
in;
and
a
defendant's
confession
to
an
entirely
unrelated
crime
committed
years
earlier
would
be
admissible
without
more.
These
are
some
of
the
absurdities
of
literalism
that
show
that
Congress could
not
have
been
writing
in
a
literalistic
frame
of
mind.94
C.
Purposes
of
the Analysis
Lawyers
and
mathematicians
use
written
analysis
for
similar
reasons.
Both
legal
analysis
and
mathematical
analysis
help
with
the
writer's
thought
process,
help
convince
the
reader
that
the
writer
is
correct,
help
expand
the
body
of
knowledge
in
the
field, and
help
educate
the
next
generation.
1.
Thinking
The
process
of
writing
a
proof
can
help the
mathematician
think
through
the
problem:
"Every
mathematician
knows
that
when
he/she writes
out
a
proof,
new
insights,
ideas,
and
questions
emerge."
95
At
the
beginning
of
the process,
the
writer
has
a
conjecture
as
to what
the
answer
should
be.
Only
by
working
through
a
proof
and finding
support
for each
step
of
the
analysis
can
the
mathematician
actually prove
that
result.
This
process
of
finding
a
method
of
proving
something
is
not
necessarily
linear
or
easy.
96
One
path
that
seems
fruitful
may
ultimately turn
out to
be
a
dead
end.
The
person
working
on
the
92.
556
U.S.
303
(2009).
93.
See
id.
at
317.
The
issue
in
Corley
was
whether
petitioner's
written
confession
to
armed
bank
robbery
and conspiracy
to
commit armed
bank
robbery
was
admissible
under
18
U.S.C.
§
3501,
despite
the
fact
he
was
held
for
an
unnecessary
and
unreasonable
period
of
time
before
being
taken
in
front
of
a
magistrate
judge.
See
id
at
312-14;
see
also
18
U.S.C.
§
3501(a)-(c)
(2018)
(mandating
confessions
admissible when
voluntary,
and
not
inadmissible
solely
because
of
delay).
94.
Corley,
556
U.S.
at
317
(declining
to
read
statute
literally).
95.
See Joseph
Auslander,
On
the
Roles
of
Proof
in Mathematics,
in
PROOF
AND
OTHER
DILEMMAS:
MATHEMATICS
AND
PHILOSOPHY
61,
66
(Roger
A.
Simons
&
Bonnie
Gold
eds.,
2011);
see
also
Claudio
Bemardi,
What
Mathematical
Logic
Says
About
the
Foundation
of
Mathematics,
in
FROM
A
HEURISTIC
POINT
OF
VIEW:
ESSAYS
IN
HONOUR
OF
CARLO
CELLUCCI
41,
44
n.2
(Cesare Cozzo
&
Emiliano
Ippoliti
eds.,
2014)
("very
often
...
proofis]
allow[]
for
...
deeper
understanding
of...
subject[s]").
96.
See
Bernardi,
supra
note
95,
at
44
(noting
difficulties
encountered
by
mathematicians).
"[I]n
mathematical
experience,
when checking
a
method,
testing
a
tool,
or
hoping that
an
application
will
follow,
there
are
very
often
trials
and
failures."
Id.
22
[Vol.
LII:5
2019]
CONAECTIONSBETWEENLEGAL
AND
MATEMATICAL
REASONIWG
proof
then
has
to
back
up
and try
a
different
tactic,
continuing
to
reason
through
the
problem,
but
now
in
a
different
way. While
trying
to
fill
the
gap
in
his
proof
of
Fermat's
Last
Theorem,
Wiles
collaborated
with his
former
student,
Richard
Taylor.
97
Wiles
became
convinced
that
the
method
they
were
trying
to
use
would
not
work,
but
Taylor
was
not
yet
certain.
98
As
Wiles
thought
about
why
the
method
was not
working, he suddenly realized that
"what
was
making
it
not
work
was
exactly
what would
make
a
method
[he
had]
tried
three
years before
work."
99
By
continuing to
try
to
work
through
the
problem,
Wiles
was
able
to
gain
insight
into the
solution
in
a
way
he
could
never
have
predicted
at
the
outset.
For lawyers, too,
writing
is
thinking.1
00
The
process
of
writing
out
a
legal
argument
can
help
the
student-or
the
practitioner or
adjudicator-think
through
the
legal
problem,
grapple
with
any
weaknesses, and arrive
at
a
solution.
While
a
legal
writer may
have an
intuitive
sense
of
the
"right"
answer,
the
process
of
articulating
the
argument
thoroughly and
carefully
can
help the
writer
either
confirm
that
answer
or
determine
that the
result
should
be
different.
When
the
writer
is
an advocate who
must
argue
for
a
particular
result, the
writing
process
can
help
identify
weak points and
claims
that
will
not work
so
that
the
advocate
can
rebut
counterarguments,
shore
up
stronger
arguments,
and,
if
necessary,
concede
claims
and
arguments
that
are
not
supportable.
2.
Convincing
Another
purpose
of
a
mathematical
proof
is
to
convince the
reader that
the
analysis
is
correct.
101
If
the
writer
has
shown
the
reader
how known
rules apply
to the
givens,
using
sound
reasoning
at
each
step
of
the
proof, then
the
reader
should
agree
with
the
writer that
the
theorem
has
been
proven.1
02
When
Wiles
completed his
proof
of
Fermat's
Last
Theorem, he
had
to
use
the
proof
to convince
his readers
that
each
step
of
the
proof
was
correct
and that,
therefore,
the
theorem
was true.
03
During
the
peer
review
process,
the
referees
97.
See Kolata,
supra
note
14
(describing
Wiles's work
with
Taylor).
98.
See
id.
99.
Id.
100.
See,
e.g., George
D.
Gopen,
CCISSR:
The
Perfect
Way
to
Teach
Legal
Writing,
13
LEGAL
WRITING:
J.
LEGAL
WRITING
INST.
315,
322
(2007)
("Writing
is
thinking;
thinking
is
writing.");
Roger
J.
Traynor,
Some Open
Questions
on
the
Work
ofState
Appellate
Courts,
24
U. CHI. L.
REV.
211, 218
(1957)
("I
have
not found
a
better
test
for
the solution
of
a
case than
its
articulation
in
writing,
which is
thinking
at
its
hardest.");
Joseph Kimble,
On
Legal-
Writing
Programs,
PERSPS.:
TEACHING LEGAL
RES.
&
WRITING,
Winter/Spring,
1994,
at
44 WL
2
No.
2/3
PERSPEC
43
("[W]riting
is
thinking.
Thinking
on
paper.
Thinking
made
visible.").
101.
See
HAMMACK,
supra
note
28,
at
89;
see
also
CLAuDI
ALsINA
&
ROGER
B.
NELSEN,
CHARMING
PROOFS:
A
JOURNEY INTO
ELEGANT
MATHEMATICS,
at
xix
(2010).
"A
proof
of
a
theorem
should
be
absolutely
convincing."
See
HAMMACK,
supra
note
28,
at
89.
"[A]
proof
is
an
argument
to
convince the
reader
that
a
mathematical
statement must
be
true."
ALSINA
&
NELSON,
supra,
at
xix.
102.
See
REUBEN
HERSH, WHAT
IS
MATHEMATICS,
REALLY?
63
(1997).
"Practical
mathematical
proof
is
what
we
do to
make
each other believe
our
theorems.
It's
argument
that
convinces the
qualified,
skeptical
expert."
Id.
103.
See
SINGH,
supra
note
2,
at
33
(reiterating
need for
rigorous
independent review
of
Wiles's
proof).
23
SUFFOLK
UNIVERSITYLAWREVIEW
periodically
asked
Wiles
for
further
explanation
when
they
came
across
portions
of
the
proof
that
they
did
not
understand
or that
they
felt
were
incomplete.
104
Each
time,
after
further
discussion
with
Wiles,
the
referees
were
able
to
get
enough
clarification
to
move
on
with
their
review
of
the
proof.
105
Until,
one
day,
Wiles's
explanation
of
the
reasons
behind
a
certain
assertion
was
not
convincing
enough.1
0
6
Although
both
Wiles
and
the
referee
believed
the
assertion
to
be
true,
Wiles
realized
his
proof
was
incomplete
and
he
went
back
to
the
drawing
board
to
bridge
the
gap,
to
make
sure
the entire
proof
was
convincing.
0 7
In legal
writing,
too, the
writer
is
trying
to
convince
the
reader
that
the
analysis
is
correct.
Most
introductory
legal
writing
texts
and
courses
contain
a
basic
divide
between
predictive
and
persuasive
legal writing.
In
predictive
writing-
exemplified
by
the
office
memorandum-the
writer
attempts
to analyze
the
law
objectively
and
predict
the
most
likely
outcome
based
on
the
facts
of
the
case
and
the
existing
law.
0
8
In
persuasive
writing-exemplified
by
the
trial
or
appellate
brief-the
writer
attempts
to
advocate
for
the outcome
that
will
lead
to
the
best
result
for
the
client.1
09
Considering "persuasion"
more
broadly,
however,
both
types
of
legal
writing
are
persuasive.
Even
with
the
objective
memo,
the
writer
wants
to
convince
the
reader
that
the
analysis
is
correct.
1
o
If
the
writer
has
presented
a
complete
picture
of
the
facts
and
the
law,
and
shown
the
reader
how
the
law
applies
to
the
facts
using
sound
reasoning,
the
reader
should
agree
with
the
writer's
conclusions.
A
judicial
opinion,
too,
is
a
persuasive
document
in
this
way.
In
writing
an
opinion,
a
judge
seeks
to
justify
the decision
using
convincing
legal
reasoning.
3.
Expanding
Knowledge
Mathematicians
also
use
proofs
to
expand
the
realm
of
mathematical
knowledge
by
establishing
the
truth
of
previously
unproven
mathematical
assertions.
Though
a
mathematician
may
feel
a
hypothesis
is
likely
true,
until
there
is
a
valid
proof
of
that
hypothesis,
the
mathematician
cannot
rely
on
the
hypothesis.
Once
the
mathematician
understands
a
valid
proof
of
the hypothesis,
however,
the
mathematician
can
then
believe
in
and
rely
on the
result.
Moreover,
once
a
proof
has
been
published
in
a
peer-reviewed
journal,
then
"the
result
is
presumed
correct,
unless
there
is
a
compelling
reason
to
believe
otherwise.""'
The
new
theorem
is
incorporated
into
the
universe
of
mathematical
knowledge
104.
See
id.
at
256-57
(recounting
email
exchanges
between
referees
and Wiles).
105.
See
id.
at
256.
106.
See
id.
at
259
(describing
referee's
persistence
in asking
Wiles
for
further
explanation).
107.
See
SINGH,
supra
note
2,
at
257-58
(relating
discovery
of
error).
108.
See,
e.g.,
SHAPo
Ir
A..,
supra
note
44, at
149
(claiming
legal
memorandum
"one
of..
.
most
common
forms
of
legal
writing").
109.
See
id.
at
319.
110.
See
id. at
149
(explaining
purpose
of
objective
memorandum).
111.
See
Auslander,
supra
note
95,
at
65.
[Vol.
LII:5
24
2019]
CONNECTIONS
BETWEEN
LEGAL
AND
MATHEMATICAL
REASONING
that
can
be
used
and
built
upon
by
mathematicians
the
world
over,
even
by
mathematicians
who
have
not
personally
read and
understood
the
proof.
1 12
The
proof
of
Fermat's
Last
Theorem
was
complex-so
complex
that
it
took
months
to validate
and
only
a
handful
of
mathematicians
had
the
right
combination
of
specialized
knowledge
to
understand
the
proof.
Nonetheless,
once
the
proof
was
validated
and
published,
mathematicians
accepted
the
result.
While
the
proof
is
complex,
the
theorem
itself
is
simple:
For
any
positive
integer
n
greater
than
2,
there
are
no
whole
numbers
a,
b,
and
c
that
make
the
equation
true
a"
+
bn
=
cn.
Even
someone
who
is
not
able
to
understand
the
proof
of
Fermat's
Last
Theorem
can
understand
the
theorem
itself
and
can
therefore
use
the
theorem
in
other
proofs.
Publication
by
peer
review
is,
of
course,
not
a
fool-proof
method
of
determining
the
validity
of
a
proof.
Incorrect
proofs
of
valid
results-and
incorrect
proofs
of
incorrect
results-have
been
published
and
relied
on.'
Furthermore,
even
when
the
proof
and
its
result
are
correct,
"applying
a
result
mechanically,
without
an
understanding
of
the
proof,
can
lead
to
errors."
1 14
in
fact,
the
problem
with
Wiles's
proof
of
Fermat's
Last
Theorem-the
gap
it
took
him
a
year
to
fill-occurred
in
a
portion
of
the
proof
where
he
relied
on
a
colleague's
method
that
he
"didn't
feel
completely
comfortable
with.""'
The
analogy
here
in
the
field
of
law
is
not
to
memos
and
briefs
but
rather
to
judicial
opinions.
Just
as
publication
of
a
proof
allows
mathematicians
to
make
use
of
the result,
once
a
court
has
ruled
on
a
case
and
issued
an
opinion,
the
holding
is
incorporated
into
the
universe
of
legal
knowledge
and lawyers
can
rely
on
the
opinion
to
make
new
arguments.
As
with
a
peer-reviewed
and
published
proof,
precautions
apply.
Courts
can-and
do-make
mistakes,
so
the
law
as
stated
in
a
court's
opinion
is
not
necessarily
correct.
Furthermore,
as
time
passes,
a
"correct"
result
may
be
overturned
by
a
later
court.
Lawyers
must
also
be
careful
to
understand
a
court's
reasoning
before
attempting
to
use
language
from
an
opinion.
Just
as
"applying
a
[mathematical]
result
mechanically,
without
an
understanding
of
the proof,
can
lead
to errors,"
so
too
can
applying
a
court's
holding
mechanically
lead
to
mistakes
by
a
lawyer.
1
1
6
4.
Teaching
In
mathematics
courses,
students
engage
in
proof
as
part
of
the
learning
process.
Mathematics
students,
even
at the
college
level,
are
not
expected
to
112.
See
id.
(explaining
certification
of
proofs
"allows
[mathematicians]
to
use
it
in
further
research").
113.
See
id.;
Reuben
Hersh,
To
Establish
New
Mathematics,
We
Use
Mental
Models
and
Build
on
Established
Mathematics,
in
FROM
A
HEURISTIC
POINT
OF
VIEW:
ESSAYS
IN
HONOUR
OF
CARLO
CELLUCCI,
supra
note
95,
at
127,
137
(recounting
examples
of
mistakes
needing
correction).
114.
Auslander,
supra
note
95,
at
65.
"[A]
mathematician
is
not
absolved
from
understanding
the
proof,
even
when
the
result
in
question
has
been
accepted
by
the
mathematical
community."
Id.
115.
See
Kolata,
supra
note
14
(recounting
struggle
to
solve
Fermat's
Last
Theorem).
116.
See
Auslander,
supra
note
95,
at
65
(referring
to proofs
in
mathematics).
25
SUFFOLK
UNIVERSITYLAWREVIEW
establish
new
theorems
never
before
proven.
Instead,
they
prove
theorems
that
have
been
understood
by
mathematicians
for
hundreds
or
even
thousands
of
years.
The
purpose
of
this
is
not
for
the
end
result
of
the
proof,
but
for
the
pedagogical
value
of
the
process.
The
reasoning
process-moving
step
by
step
through
the
syllogisms
to
reach
a
solution
to
the
problem
at
hand-is
critical
to
engaging
in
upper-level
mathematics.
1 1
7
"Theorems
and
their
proofs
lie
at
the
heart
of
mathematics."
1
18
Thus,
when
students
learn
mathematical
proof,
they
are
improving
their
understanding
of
mathematics
and
learning
to
think
like
a
mathematician.
1
19
Similarly,
it
is
a
clich6
that
law
school
teaches
students
to
"think
like
a
lawyer."
1
20
IRAC
is
a
large
part
of
that.
Identifying
legal
issues,
knowing
and
understanding
legal
rules,
and
being
able
to
apply
those
rules
to
facts
are
all
important
pieces
of
"thinking
like
a
lawyer."
1 2
1
III.
BuT
DOESN'T
LAw
HAVE
LESS
CERTAINTY
THAN
MATHEMATICS?
"[T]he
moment
you
leave
the
path
of
merely
logical
deduction
you
lose
the
illusion
of
certainty
which
makes
legal
reasoning
seem
like
mathematics.
But
the
certainty
is
only
an
illusion,
nevertheless."
22
"Absolute
certainty
is
what
many
yearn
for
in
childhood,
but
learn
to
live
without
in adult
life,
including
in
mathematics."
23
Comparisons
between
legal
reasoning
and
mathematical
reasoning
are
not
new,
nor
are
criticisms
of
such
comparisons.
1 24
Langdell
famously
wanted
to
treat
law
as
a
science
and
developed
his
casebook
method
at
Harvard
around
that
117.
See
John
H.
Conway,
Foreword
to
How
To
SOLVE
IT:
A
NEW
ASPECT
OF
MATHEMATICAL
METHOD,
supra
note
66,
at
xx.
"Mathematics,
you
see,
is
not
a
spectator
sport.
To
understand
mathematics
means
to
be
able
to
do
mathematics.
And
what
does
it
mean
[to
be]
doing
mathematics?
In
the
first
place,
it
means
to
be
able
to
solve
mathematical
problems."
Id.
(quoting
George
Polya).
118.
See
ALSINA
&
NELSON,
supra
note
101,
at
ix.
119.
See
WILLIAM
BYERS,
DEEP
THINKING:
WHAT
MATHEMATICS
CAN TEACH
US
ABOUT
THE
MIND
106
(2015)
("The
formal
theorems
and proofs
deepen
our
understanding
of
the concepts
involved.
Conversely
a
deeper
conceptual
understanding
leads
to
new
and
better
theorems.").
120.
See
TISCIONE,
supra
note
57,
at
1
(noting
common
analytical
thinking
development
among
law
students).
121.
See
id.
122.
Oliver
Wendell
Holmes,
Jr.,
Privilege,
Malice,
and
Intent,
8
HARv.
L.
REV.
1,
7
(1894).
123.
RUEBEN
HERSH,
EXPERIENCING
MATHEMATICS:
WHAT
Do
WE
Do,
WHEN
WE
Do
MATHEMATICS?
81
(2014).
124.
See,
e.g., Morris
R.
Cohen,
The
Place
ofLogic
in
the
Law,
29
HARV.
L.
REV.
622,
624
(1916).
We try
to
reduce
the law
to
the
smallest
number
of
general
principles
from
which
all
possible
cases
can
be
reached,
just
as
we
try
to
reduce
our
knowledge
of
nature
to
a
deductive
mathematical
system....
The law,
of
course,
never
succeeds
in
becoming
a
completely
deductive
system.
It
does
not
even
succeed
in
becoming
completely
consistent.
Id.
26
[Vol.
LII:5
2019] CONNECTIONSBETWEENLEGAL
AMD
MATHIEMATICAL
REASONIVG
idea.
125
Holmes
agreed
that
law
was
a
science,
126
but
maintained
that
law and
mathematics cannot
be
equated.1
27
When
the
attempts
failed, the
resulting
insights into
the
nature
of
mathematics
were
truly
groundbreaking. The first
insight
came
in
the
area
of
geometry.
For
more
than
a
thousand
years,
following
in
Euclid's
footsteps, mathematicians
studied
geometry
based
on
five
postulates:
1.
For
any
two points,
a
straight
line
segment
can
be
drawn
to
join
those
two
points.
2.
Any
straight
line
segment
can
be
extended
indefinitely
in
a
straight
line.
3.
For
any
straight
line
segment,
a
circle
can be
drawn
having
the
segment
as
radius
and one
endpoint
as
center.
4.
All
right
angles
are
congruent.
5.
If
two straight
lines
intersect
a
third
straight
line
in
such
a
way
that
the
sum
of
the
inner
angles on
one
side
of
the intersection
line
is
less
than
two
right
angles,
then
the
two
lines
must
intersect
each
other
on
that
side
if
extended
indefinitely.
12
8
In his
Elements,
Euclid
built on
these
five
postulates
to
prove theorems
and
explain
geometry.1
29
The
first
four statements seemed
self-evident
and
therefore
were
properly
considered
postulates
because
they
were true
but could not
be
proven.
13 0
The
fifth
postulate,
however,
caused
some
trouble
because
of
a
sense
that
it
was not
as
self-evident
as
the
other
postulates and therefore should
be
proven
rather
than
merely assumed
to
be
true.
Euclid
was
"reluctant"
to
introduce
it,
and
mathematicians
who
followed
Euclid
attempted
to
prove
the
fifth
postulate,
all
without
success.'
31
The
insight,
when
it
eventually
came
many centuries
after
Euclid,
was
that
the
fifth
postulate
is
not
necessarily
true.
One
method
mathematicians
tried
to
use
to prove
the
fifth
postulate
was
proof
by
contradiction.
32
Though
a
fruitless
effort,
one
of
those mathematicians
who
attempted
it,
instead
of
giving
up
125.
See,
e.g.,
Gerald
P.
Lopez,
Transform-Don't
Just
Tinker
with-Legal
Education,
23
CLINICAL
L.
REV.
471,
517
(2017)
(describing
Langell's
teaching
methods).
126.
Oliver
Wendell
Holmes,
Law
in
Science
and
Science
in
Law,
12
HARV.
L.
REV.
443,
452
(1899).
"The
true
science
of
the law does
not
consist mainly
in
a
theological
working
out
of
dogma
or a
logical
development
as
in
mathematics
.
.
.;
an
even
more
important part
consists
in
the
establishment
of
its
postulates
from
within
upon accurately
measured
social
desires instead
of
tradition."
Id.
127.
See
Oliver
Wendell
Holmes,
The
Path
of
the
Law,
10
HARV.
L.
REv.
457,
465
(1897).
"The
danger
of
which
I
speak
is
...
the
notion
that
a given
system,
ours,
for
instance, can
be
worked
out
like
mathematics from
some
general
axioms
of
conduct."
Id.
128.
See
COXETER,
supra
note
21,
at
1
(listing
Euclid's
five
postulates).
129.
See
Gray,
supra
note
21,
at
83-84
(describing organization
of
Euclid's
Elements).
130.
See
COXETER,
supra
note
21,
at
2
(contrasting
first
four
postulates
with
fifth
because
"not
self-evident
like
.
..
others").
131.
See
id.
(explaining distinction
between postulate
five
and
other
four postulates).
132.
See
MEYER,
supra
note
31,
at
106-07.
27
SUFFOLK
UNIVERSITYL4
WREVIEW
because
he
failed
to
prove
the
fifth
postulate,
kept
delving
deeper
into
the
insights
he
gained
when
he
assumed
that
the
fifth
postulate
was
false.'
33
In
doing
so,
he-along
with
a
few
other
mathematicians
who
had
similar
insights
in
the
same
time
period--discovered
hyperbolic
geometry,
which
is
now
considered
one
of
two
main
types
of
non-Euclidean
geometry.'
3 4
While
the
fifth
postulate
is
true
of
lines
and points
on
a
flat
plane,
it
is
not
true
in
hyperbolic
geometry,
which
is
the
geometry
of
a
saddle-shaped
surface.
For
thousands
of
years,
mathematicians
assumed
that
Euclid's
postulates
were
true.
But
with
the
discovery
of
non-Euclidean
geometry,
it
became
clear
that
they
may
or
may
not
be true
depending
on the
system
in
which
one
was
operating.
Furthermore,
different
systems
can
be
built
upon
different
axioms.
Mathematicians
developed
hyperbolic
geometry
based
on
a
different
set
of
axioms,
only
some
of
which
are
the
same
as
Euclid's
postulates.'
3 5
Elliptic
geometry,
the
other
main
type
of
non-Euclidean
geometry,
is
built
on
yet
another
set
of
axioms.'
3 6
In
the
early
1900s,
mathematician
David
Hilbert
was
working
on
an
axiomatization
of
mathematics.
1 37
That
is,
he
wanted
to
develop
formal
systems
of
mathematics
starting
from
a
small
number
of
basic
axioms,
much
as
geometry
is
built
upon
a
foundation
of
axioms.
3
8
Importantly,
Hilbert,
who
is
often
described
as
a
formalist,
wanted
to
show
that
each
formal
system
was
complete
and
consistent.
139
A
system
is
complete
if
any
statement
within
that
system
can
be
proven
to
be
either
true
or
false.1
40
A
system
is
consistent
if,
using
the
axioms
of
that
system,
it
is
not
possible
to
prove
a
statement
both
true
and
false.'
41
While
Hilbert
was
attempting
to
develop
these
formal
systems,
another
mathematician
announced
a
proof
that
suggested
Hilbert
would
never
succeed.
Kurt
Godel,
a
young
mathematician
who
had
only
recently
finished
his
dissertation,
wanted
to
support
Hilbert's
goals
by
proving
the
consistency
of
a
subset
of
mathematics
known
as
analysis.
142
Instead,
G6del's
reasoning
led
to
133.
See
id.
at
107;
see
also
CoXETERsupra
note
21,
at
5.
134.
See
COXETER,
supra
note
21,
at
4-5
(describing
development
of
non-Euclidean
hyperbolic
geometry
through
attempts
to
prove
parallel
postulate).
Coxeter
noted
"non-Euclidean
geometry"
typically
refers
to
hyperbolic
and elliptic
geometry.
Id.
135.
See
MEYER,
supra
note
31,
at
102
(defining
hyperbolic
geometry).
136.
See
id.
at
95
(defining
elliptic
geometry).
137.
See
Donald
Gillies,
Serendipity
and
Mathematical
Logic,
in
FROM
A
HEURISTIC
POINT
OF
VIEW:
ESSAYS
IN
HONOUR
OF
CARLO
CELLUCCI,
supra
note
95,
at
23,
25.
In
Hilbert's
version
of
formalism,
"mathematics
consisted
of
a
collection
of
formal
systems
in
each
of
which
the
theorems
were
deduced
from
the
axioms
using
mathematical
logic."
Id.
138.
See
id.
139.
See
id.;
see
also
Stephen
C.
Kleene,
Introductory
Note
to
1930b,
1931
and
1932b,
in
1
KURT
GODEL:
COLLECTED
WORKS
126, 127
(Solomon
Feferman
et
al.
eds.,
1986)
(discussing
Hilbert's
proposals).
140.
See
RON
AHARONI,
MATHEMATICS,
POETRY
AND
BEAUTY
135
(2015)
(describing
Hilbert's
goal
of
proving
completeness
for system
of
axioms);
Kleene,
supra
note
139,
at
127.
141.
See
AHARONI,
supra
note
140, at
135
(describing
Hilbert's
goal
of
proving
consistency
for
system
of
axioms);
see
also
Kleene,
supra
note
139,
at
127
(explaining
Hilbert's
consistency
of
systems
requirement).
142.
See
Kleene,
supra
note
139,
at
127.
[Vol.
LH:5
28
2019]
CONAECTIONSBETWEENLEGAL
AND
MATHEMATICAL
REASONDVG
his
Incompleteness
Theorem,
which
called
into question
the
attempts
at
mathematical
formalism.'
43
Godel
was
inspired
in part
by
the
ancient
Liar's
Paradox,
1
"
which
is
the
paradox
that
results
from
trying
to decide
whether
the
following
sentence
is
true
or
false:
This
sentence
is
false.
If
the
statement "this
sentence
is
false"
is
true,
then
by
its
own
terms,
it
is
false.
And
if
the
statement
"this
sentence
is
false"
is false,
then,
because it says it
is
false,
it
must
be
true.
Either route leads
to
a
contradiction
and
it
is
not
possible
to
call
the
sentence
either true
or
false.
G6del's
proof
extended
this
idea
to
the concept
of
provability.
145
G6del
proved
that
if
a
formal
system
of
mathematics is
consistent,
then
there
is
a
statement
within that
system
that
is
not provable,
meaning that
there
is
no
proof
that
the
statement
is
true
and
also
no
proof
that
the
statement
is
false.
146
"Gdel's
incompleteness
theorem
implies
that
there
can
be no
formal
system
that
is
consistent,
yet
powerful
enough to
serve
as
a
basis
for
all
of
the
mathematics
that
we
do."
147
These
insights
do
not
change
the
fact
that
mathematical
reasoning
and
mathematical
proof
are
useful.
Mathematicians
continue
to
theorize about
mathematics and prove
theorems, though
they
now
do
so
with
an
awareness
that
the axioms
on
which
the
theorems
are
built
are
merely
assumptions-a
working
model for
one
view
of
a
mathematical
universe-and
that
the model
itself
cannot
be
complete.
148
Similarly,
in
the
law,
good,
solid
reasoning
is
useful even
though
lawyers
should
be aware
that
the
system
of
laws
is
never
complete.
In
attempting
to describe
commonalities
among legal
realist
thinkers
of
his
day,
Llewellyn identified
several
points, including
the
following:
a
"conception
of
law
in
flux";
a
"conception
of
society
in
flux
...
so
that
the
probability
is
always
given
that
any
portion
of
law
needs
reexamination
to determine
how far
it
fits
the
society it
purports
to
serve";
a
"[d]istrust
of
traditional
legal
rules and
concepts
insofar
as
they
purport
to
describe
what
either courts
or
people
are
actually
doing";
and
a
"belief
in
the
worthwhileness
of
grouping
cases
and
legal
143.
See
id.
at
127-28.
144.
See
Kurt G6del, On
Formally
Undecidable
Propositions
ofPrincipia
Mathematica
and
Related
Systems
I,
in
1
KURT
GODEL:
COLLECTED
WORKS,
supra
note
139,
145,
149-5
1.
145.
See
id.
(laying out
"a
proposition that
says
about
itself
that
it
is
not
provable").
146.
See
Bernardi,
supra
note 95,
at
26
(summarizing
the
result
of
Gbdel's
First Incompleteness Theorem).
147.
William
P.
Thurston,
On
Proof
and
Progress
in
Mathematics,
30
BULL.
AM.
MATHEMATICAL
Soc'Y.
161,
170
(1994).
148.
See
RICHARD
L.
EPSTEIN, REASONING
IN
SCIENCE
AND
MATHEMATICS:
ESSAYS
ON LOGIC
AS
THE
ART
OF
REASONING
WELL
77
(2012) (highlighting assumptions
underlying
mathematical proofs).
Some
philosophers
of
mathematics
even
argue that proofs built
upon
axioms
do
not truly
prove
anything
since
the
axioms
themselves
"are
not
established,
they
are
simply
postulated."
Hersh,
supra
note
113,
at
134.
"Perhaps
in
the
nineteenth
century,
logic
was
regarded
as
a
way to guarantee
the
certainty
of
mathematics. But nowadays
we
do
not
expect
that
much:
[I]t
seems
naive,
and
perhaps
even
futile,
to hope
for
a
definitive,
proven
certainty
of
mathematics."
Bernardi,
supra
note
95,
at 42.
29
SUFFOLK
UITAVERSITYLA
WREVIEW
situations
into
narrower
categories
than
has
been
the
practice
in
the
past."
1 49
All
of
these
points
reveal
similarities
between
law
and
mathematics
rather
than
differences.
Beginning
with
the
last
point,
"grouping
cases
and
legal
situations
into
narrower
categories,"
i.e.,
creating
narrower
rules,
does
not
change
the
reasoning
process
necessary
to
apply
rules-whether
broad
or
narrow-to
the
facts
at
hand.
Indeed,
Llewellyn
thought
logic
had
a
place
in
legal
reasoning,
though
he
believed
that
other
aspects
were
important
as
well.
150
Llewellyn
also
noted
that
legal
realists
conceive
of
both
law
and
the
society
to
which
law
should
respond
as
"in
flux."
151
Similarly,
mathematicians
now
acknowledge
that
the
field
of
mathematics
changes
and
that
it
is
a
function
of
its
time and
place.
1 52
As
philosopher
of
mathematics
Reuben
Hersh
has
observed,
"[t]he
body
of
established
mathematics
is
not
a
fixed
or
static
set
of
statements..
..
Established
mathematics
is
established
on
the
basis
of
history,
social
practice,
and
internal
coherence.
.
. .
What
has
been
published
remains
subject
to
criticism
or
correction."
1 53
The
legal
realists'
"[d]istrust
of
traditional
legal
rules
and
concepts
insofar
as
they
purport
to
describe
what
either
courts
or
people
are
actually
doing,"
1 54
also
has
echoes
in
mathematics.
"A
mathematical
theorem
does
not
show
that
a
claim
is
true.
It
shows
that
the
claim
follows
from
the
assumptions
of
the
theory."
155
The
Greek
word
for
"axiom"
originated
not
in
mathematics,
but
in
rhetoric.
1 56
Originally,
it
meant
a
statement
that
was
assumed
to
be
true
for
purposes
of
the
149.
See
Karl
N.
Llewellyn,
Some
Realism
about
Realism-Responding
to
Dean
Pound,
44
HARV.
L.
REV.
1222,
1235-37
(1931)
(describing
"common
points
of
departure"
of
those
in
legal
realism
movement).
150.
See
Karl
N.
Llewellyn,
A
Realistic
Jurisprudence-The
Next
Step,
30
COLUM.
L.
REV.
431,
447
n.12
(1930).
A
careful
study
of
the formal
logic
ofjudicial
opinions
would
be
a
useful
study.
But
I
would
urge
that
even
its
usefulness
would
be
hugely
increased
by
an
equally
careful
study
of
the
instrumentalism,
the
pragmatic
and
socio-psychological
decision elements
in
the
same
cases.
And
that
an
equally
geometric
increase in
illumination
would
follow
a
further careful
study
of
the
effects on the
society
concerned
of
the
same
cases.
Id.
151.
See
Llewellyn,
supra
note
149,
at
1236.
152.
See
Michael
D.
Resnick,
Proofas
a
Source
of
Truth, in
PROOF
AND
KNOWLEDGE
IN
MATHEMATICS
1,
7
(Michael
Detlefsen
ed.,
1992)
(considering
"proofs
.
. .
social
and
cultural
objects").
Proofs
"evolve
in
a
particular
social and cultural context,
and
they
have intended
audiences."
Id.
153.
See
Hersh,
supra
note
113,
at
132.
"A
warrant
is
a
justification
to
act
on
an
assertion,
a
justification
based
on
lived
experience."
Id.
at
137.
154.
See
Llewellyn,
supra
note
149,
at
1237.
155.
See
EPSTEIN,
supra
note
148,
at
77;
see
also
Thurston,
supra
note
147,
at
170
("[m]ost
mathematicians
adhere
to
foundational
principles
.. .
known
to
be
polite
fictions.").
156.
See Hans
Niels
Jahnke,
The
Conjoint
Origin
ofProofand
Theoretical
Physics,
in
THE BEST
WRITING
ON
MATHEMATICS
2011,
at
236,
237-38
(Mircea
Pitici
ed.,
2012)
(detailing
linguistic
history
of
mathematical
terms).
[Vol.
LII:5
30
2019]
CONNECTIONSBETWEENLEGAL
AND
MATEMATICAL
REASONING
rhetorical
argument.
15
7
The
absolute
truth
of
the
axiom
was
not
necessary,
just
that
those
arguing
agreed
to
take
the
statement
as
true.
1
s
5
s
Around
the same
time
the
term was
imported
into
mathematics,
it
came
to
have
its
current
meaning:
a
statement
that mathematicians
believe
to
be
true
but
that
cannot
be
proved.1
59
With
the
insights
of
non-Euclidean
geometry
and
Gbdel's
Incompleteness
Theorem,
mathematicians'
understanding
of
the
word
"axiom"
is
closer
to
its
original meaning:
something
that
we
take
as
true
for
the
sake
of
argument.
160
in
the
legal
field,
lawyers
treat
rules
in
much
the
same
way.
Lawyers
take
the
laws
and rules
to
be
"true"
and
form
their
arguments
based
on
those
rules.
IV.
CAN
ANYTHING
AccOUNT
FOR
THE
SIMLARITIES?
The
work
and
theories
of
cognitive
psychologists
and
other
cognitive
scientists
may
help
explain
the
similarities
between
mathematical
analysis
and
legal
analysis.
Many
of
the
shared features
help
improve
the "fluency"
of
the
writing.
With
both
mathematical
proofs
and
legal
writing,
communication
is
an
important
function
of
the
document.
For
the
document
to
serve
its
purpose,
the
reader
of
the
document
must
understand
it
and
believe
it.
The
"fluency"
of
the
document
is
key.
Fluency
is
"a
subjective
experience
of
ease
or
difficulty
associated
with
a
mental
process.
In
other
words,
fluency
isn't
the
process
itself
but, rather,
information
about how
efficient
or
easy
that process
feels."
161
Fluent
documents,
to
the reader,
seem
to
be
easier
to
process than
disfluent
documents.
Importantly,
for
mathematical
analysis
and
for
legal
analysis,
when
a
document
is
more
fluent,
the reader
is
more
likely to
perceive
the
document
as
true
and
to
have
higher
confidence
in
the document.
162
Since
legal writers
and
157.
See
id.
at
238
(explaining
origin
of
"axiom"
terminology).
158.
See
id.
at
238-39.
159.
See
id.
(describing
shift
to
current meaning
of
axiom).
160.
See
Jahnke,
supra
note
156,
at
247-48.
[Until
the
end
of
the
nineteenth
century, mathematicians
were
convinced
that mathematics
rest
on
intuitively
secure
intrinsic
hypotheses which
determined
the
inner
identity
of
mathematics....
Then,
non-Euclidian
geometries were
discovered.
The
subsequent
discussions
about
the
foundations
of
mathematics
at
the
beginning
of
the
twentieth
century resulted
in
the
decisive
insight
that pure
mathematics
cannot exist
without
hypotheses
(axioms)
which
can
only
be
justified
extrinsically.
Id.
161.
See
Daniel
M.
Oppenheimer,
The
Secret
Life
ofFluency,
12
TRENDs
COGNITIVE
SCI.
237, 238
(2008)
(explaining role
of
fluency
in
judgments
and
cognition).
162.
See
Daniel
M.
Oppenheimer,
Consequences
ofErudite
Vernacular
Utilized
Irrespective
ofNecessity:
Problems
with
Using
Long
Words
Needlessly,
20
APPLIED
COGNITIVE
PSYCHOL.
139,
140
(2006)
(explaining
processing
"[f]luency
leads
to
higher
[j]udgements
of
truth,
confidence,
. . .
and
even
liking");
see
also
Julie
A.
Baker,
And
the
Winner
Is:
How
Principles
ofCognitive
Science
Resolve
the
Plain
Language
Debate,
80
UMKC
L.
REV.
287,
288
(2011).
Research shows that
"the
more
'fluent'
a
piece
of
written information
is,
the
better
a
reader
will
understand
it,
and
the
better
he
or
she
will
like,
trust
and
believe
it."
Baker,
supra,
at 288.
31
2019]
COAWECTIONSBETWEENLEGAL
AND
MATHEMATICAL
REASONIVG
Alternatively,
the sentences
in
a
paragraph
can
be
structured
such
that
each
sentence
adds
to
the
given
information
in
the
first
sentence.'
7
'
Sentence
1:
[given
information
A]
[new
information
B]
Sentence
2:
[given
information
A]
[new
information
C]
Sentence
3:
[given
information
A]
[new
information
D]
Aside
from
processing
the
words on
the
page,
readers
of
a
legal
analysis
or
mathematical
analysis
will
have
an easier
time
following
the
argument
if
it
uses
an
inferential
rule
known
as
modus
ponens.
Modus
ponens
is
a
rule
of
inference
that
applies
when
an
argument
takes the
following
form:
"If
p,
then
q;
p,
therefore,
q."l
7 2
Using
the
terms
of
a
syllogism, the
major premise
is,
"Ifp,
then
q."l73
The
minor
premise
is
"p."
1 74
And
the
conclusion
is
"q."1
75
More
concretely,
consider
the
following
two
statements:
1.
If
the
temperature
is
below
32
degrees
Fahrenheit,
then
the water
will
freeze.
2.
The
temperature
is
below
32
degrees Fahrenheit.
Applying
modus
ponens,
one
can
infer
that
the
water
will freeze.
Based
on
studies
involving
problem
solving
using
the
modus
ponens
rule
of
inference,
"[fior
modus
ponens,
there
is
evidence
that
people:
(a)
perform
as
well-that
is,
make
inferences
in accordance
with
the
rule-on
unfamiliar
as on
familiar
material;
[and]
(b)
perform
as
well
on
abstract
as
on
concrete
material."'
The freezing
water
example
above
is
both
concrete and
familiar-
we
know
and
understand
the
freezing
point
of
water.
Subjects
in
the studies,
however,
were
able
to
apply
the
logic
of
modusponens
even
on material
that
was
abstract
or
less
familiar.'
77
Thus,
reasoning
that
relies on
modus
ponens
should
be
easy
for
the
reader
to
follow
even
when
the
reader
is
not
very
familiar
with
the
underlying material.
171.
Seeid.at209.
172.
See
Barbara
A.
Kalinowski,
Logic
Ab
Initio:
A
Functional
Approach
to
Improve
Law
Students'
Critical
Thinking
Skills,
22
LEGAL
WRITING:
J. LEGAL
WRITING
INST.
109,
130
(2018)
(explaining
conditional
syllogisms).
173.
See
id.
at
128
(describing
parts
of
syllogism).
174.
See
id.
175.
See
id.
176.
See
Edward
E.
Smith
et
al.,
The
Case
for
Rules
in
Reasoning,
16
COGNITIVE
ScI.
1,
30
(1992).
177.
See
id.
at
11,
15.
Examples
of
abstract
and
less
familiar
rules include
the
following:
"If
the
letter
is
L,
then
the
number
is
5"
and
"[i]f
she
meets her
friend,
then
she
will
go
to a
play."
Id.
Although
the friend
going
to a
play
example
is
not
nonsensical,
it
is
less
"familiar"
because
meeting
with a
friend
does
not necessarily
always
lead to going
to a
play.
Many people
meet friends
and
then
go
for coffee,
to
lunch,
to
a
movie,
or
to do
any
of
hundreds
of
other
activities.
33
SUFFOLK
UNIVERSITYIA
WREVIEW
V.
WHAT
CAN
LAWYERS
LEARN
FROM
MATHEMATICIANS?
In
light
of
the
similarities
between written
legal
analysis
and
written
mathematical
analysis,
are
there
further
insights
to
be
gleaned
from
mathematics,
especially
in
areas
where the
two
forms
of
writing
differ?
One
difference
between
the
two
forms
lies
in
how
and
where
the
facts
are
incorporated.
In
IRAC,
the
facts
of
the
case
at
hand
do
not
appear
until
the
A
section,
when
the
facts
are
analyzed
in
light
of
the
rules
and
the
precedent.'
8
Although many
forms
of
legal
writing-such
as
memos
and
briefs-call
for
a
fact
section
before
the
legal analysis
(i.e.,
before
any
IRAC),
an
IRAC-organized
analysis
does
not
incorporate
facts
until
A.1
79
The IRAC in
Figure
3
above
is
relying
on
facts
that
the
reader
likely already
knows
from
seeing
them
earlier
in
the
document.
180
Within
the
IRAC,
however,
those
facts
do
not
appear
until
the
application
section.
181
With
mathematical
proof,
on
the
other
hand,
the
givens
are
announced
at
the
beginning
of
the
proof,
serving
as
an
immediate
starting place
for
the
analysis.'
82
Specifically,
a
proof
typically
starts
with either
the
givens and
the
statement
to
be
proven,1
83
or
with
a
statement
of
the
theorem,
which
includes
both
the
givens
and
the
statement to
be
proven.1
84
In
terms
more
familiar
to
legal
writers,
the
proof
starts with
the
issue
and
the
facts.
The facts
are
then immediately
used
and
analyzed.
In
Figure
2
above,
the
given
is
that
x
and
y
are
even
integers.'
The
proof
immediately uses that
fact
to
reason
that,
based
on the
definition
of
even
integers,
there
must
be another
integer
n
such
that
x
=
2n
and
an
integer
m
such
that
y
=
2m.
Legal
writers
should
consider
whether
it
is
appropriate
to
include
a
discussion
of
the
facts before
the
A
of
IRAC.
Following
the
mathematical
model,
the
facts
could
come either
right
before
or
after
I,
the statement
of
the
issue.
Indeed, as
Professor
Diane
Kraft
discovered when
she
analyzed
the
IRAC
form
of
federal
appellate
briefs,
many
legal
writers
already
do
this.'
87
Another
difference
between
legal
analysis and mathematical
analysis
is
in
the
interplay
between
the
rules
and
the
application
of
the
rules.
In
IRAC,
all
of
the
relevant
rules
are
discussed
in
the
R
section
before
the
writer
moves
on
to
A
to
178.
See
supra
Section
I.A.2.
179.
See,
e.g.,
CALLEROS,
supra
note
42,
at
152-53, 339, 369;
SHAPO
ET
AL.,
supra
note
44,
at 339.
180.
See
supra
Figure
3.
The
analysis
set
forth
in
Figure
3
relies
on
the
facts
that
Ms.
Garcia
and
Mr.
Jordan
promised
each
other
that
they
would
be
"like
husband
and
wife"
and
that
the
seventeen
witnesses
at
trial
all
testified
that
they
considered
Ms.
Garcia and
Mr.
Jordan
to
be
husband
and
wife.
181.
See
supra
Figure
3.
182.
See
supra
Section
IIA.
1
(explaining purpose
of
givens
in
mathematical
proof).
183.
See
supra
Figure
1.
184.
See
supra
Figure
2.
185.
See
supra
note
39
and accompanying
text (defining
"even
number").
186.
See
id.
187.
See
Diane
B.
Kraft,
CREAC
in
the
Real
World,
63
CLEv.
ST.
L.
REv.
567, 585-86 (2015)
(describing
instances
when
facts
get
incorporated
into
legal
analysis).
34
[Vol.
LII:5
2019]
CONNECTIONSBETWEENLEGAL
AND
MATHEMATICAL
REASONING
apply
those
rules.'
88
In
a
mathematical
proof,
in
contrast,
the
writer
moves
back
and
forth
between
the
rules
and
the
application
throughout
the
document.
Each
step
of
the
mathematical
proof
includes
both
a
rule
and
an
application
of
the
rule.
The
analysis
continues
step-by-step
until
the
writer
reaches
the
desired
conclusion.
18 9
In IRAC
terms,
a
proof
would
look
more
like
IRA-RA-RA-RA-
RA-C
rather
than
the
standard
IRAC.
Some
of
the
difference
here
between
mathematical
proof
and
IRAC
is
due
to
the
fact
that
the
rules
in
law
are
more
complex
and
often
have
multiple
parts,
elements,
or
factors.
It
may
not
make
sense
to
separate
those
pieces
because
they
all
work
together.
Nonetheless,
legal
writers
should
consider whether
and
when
a
more
step-wise
analysis could
be appropriate.
Furthermore,
even
when
the
complexity
of
the
rules does not
allow
this
approach,
legal writers
should
use
the
"given-new"
sentence
structure
within
their
larger
R's
and
A's.'
90
As
Professors
Catherine
Cameron
and
Lance
Long
explain
in
their
book,
The
Science
Behind
the
Art
of
Legal
Writing,
"a
legal
writer can
be
guided
by
what
we
know
about
good
organization
on
the
paragraph
level-beginning
the
paragraph
with
a
topic
sentence
...
and
organizing
sentences
in
a
manner
that
takes
advantage
of
the
given-new
structure
will
be
the
most
effective
organization
for
a
reader."'
91
Legal
writers should
consider
IRAC
to
be a
flexible organizational
scheme
rather than
a
rigid
formula.
To
the
extent
that,
with
any
given
argument,
it
is
helpful
to
lay
out
the
facts
earlier
in
IRAC
or
to
move
back
and forth
between
R
and
A
multiple
times,
legal
writers
should
do
so.
VI.
CONCLUSION
Law
and
mathematics,
though
seemingly
very
different,
are
built
on
the
same
core
of
reasoning
and analysis.
With
some
understanding
of
the
methods
of
mathematical
proof,
lawyers can
harness their
inner
(possibly
dormant)
mathematicians
to
develop
solid,
convincing
analyses.
188.
see
COUGHLIN
ET
AL.,
supra
note 52,
at
83-85
(noting
"application
...
include[s]
[no]
..
.new law").
189.
See
supra
Figures
1,
2
(demonstrating
proofs
and
two-column
proofs).
190.
See
supra
notes
165-170
and
accompanying
text
(describing
importance
of
"given-new" information
structure
to fluency).
191.
CATHERINE
J.
CAMERON
&
LANCE
N.
LONG,
THE
SCIENCE
BEHIND
THE
ART
OF
LEGAL
WRITING
91
(2015).
35
Cn
HONESTAS
ET
DILIGENTIA
BOS
TON
19
0
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