We bake bread to satisfy our hunger, and we prove theorems to satisfy our hunger for knowledge and existence. Here I explore this analogy

Dexter, Cache

English

Scholarship@Claremont

Journal of Humanistic Mathematics Volume 9 | Issue 2 July 2019 How to Bake a Theorem Cache Dexter Follow this and additional works at: https://scholarship.claremont.edu/jhm  Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Dexter, C. "How to Bake a Theorem," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), pages 364-371. DOI: 10.5642/jhummath.201902.31 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss2/31 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. How to Bake a TheoremCache Dexter1American Fork, Utah, USAcachedexter@gmail.comWe bake bread to satisfy our hunger, and we prove theorems to satisfy ourhunger for knowledge and existence. Here I explore this analogy.To bake bread, all you need is flour, water, yeast, and salt. Flour to shapeand form, water to bind and activate, yeast to leaven, and salt to flavor.To bake a theorem, you also need flour, water, yeast, and salt, of sorts.A mathematical theorem is a statement about some relationship betweenmathematical objects. Importantly, a theorem must have a proof. A proof isa series of logical steps that trace a theorem back to self-evident mathematicaltruths. It isn’t enough to show that a theorem holds for a few or even manycases. A proof shows that a theorem is true all of the time. A theoremwithout a proof is just a guess, a conjecture. It is merely dough, limp anduseless until baked through. It does not satisfy our hunger for certainty.Theorems are baked with the same ingredients as bread, in a way. Mathe-matical flour is the raw material, the definitions of the mathematical objects.A triangle has three straight sides and three angles. If one of the angles of atriangle measures 90◦, then the triangle is a right triangle. The rules and pro-cedures of math are the water. These rules activate the definitions in the flourand form something kneadable and workable. To find the area of a square,multiply the length of the square by the width. The logic and the flow of themathematical argument make up the yeast, the lifting agent. If two squareshave the same side length, then the two squares have equal area. And themathematical salt is the details, the wording, the commentary on the math.1The author teaches secondary mathematics in Utah, USA, and writes about math inhis spare time.Journal of Humanistic Mathematics Volume 9 Number 2 (July 2019)Cache Dexter 365Math isn’t just pushing around symbols; it is the exploration of patterns andideas. Pointing out these patterns gives flavor to a proof.The Pythagorean theorem is an ancient loaf, proved thousands of years ago.Theorem. Given a right triangle with sides of length a, b, and c, where theside of length c is opposite the right angle, then a2 + b2 = c2.Before giving the proof of this theorem, a little aside about proofs. Sometimesreading a proof isn’t that interesting, or even worth the time. The proof ofthe classification of finite simple groups is contained in hundreds of articlesthat total over 10,000 pages. In cases like these, only a few people actuallycheck to make sure the proof is correct and the rest of us just skip to thestatement of the theorem. The statement of the theorem at the end of theproof is sometimes preceded by a little symbol ∴ , three dots in a trianglethat mean “therefore”. It’s the mathematical symbol for skipping to the endbecause the proof was too long, and you didn’t read it.However, if the proof of a theorem is not very long then reading it maybe valuable. Proofs have a way of showing how mathematics works andexplaining why the patterns are interesting. The proof of the Pythagoreantheorem is not long, but if you feel so inclined you can skip to the “∴” at theend. Here’s the proof, in a picture:aaaaaaaabbbbbbbbccccThe square on the left contains four right triangles each with sides of lengthsa, b, and c. The remaining area of the white square is c2, since the white366 How to Bake a Theoremsquare has sides of length c. The square on the right is the same size andcontains four identical right triangles, just rearranged. The remaining whitearea is formed by two squares, one with sides of length a and the other withsides of length b. The white area on the right is the sum of these two areas,which is a2 + b2.∴ a2 + b2 = c2.It doesn’t matter what the numbers a, b, and c are. We could draw the sametypes of squares starting with any right triangle. That’s what makes this aproof that the Pythagorean theorem holds for every right triangle. Drawinga right triangle and measuring its sides are the flour of this theorem. Therules of geometry are the water, allowing us to draw additional lines andshapes around the right triangle. The logic of rearranging the shapes withoutchanging their areas provides the lift, the yeast that finishes the proof thata2 + b2 does in fact equal c2. A little salt might be that this is only one ofmany ways to prove the Pythagorean theorem.The mathematician Andrew Vazsonyi recounted his first meeting with an-other mathematician, Paul Erdős, when he was 14 and Erdős was 17 [1].“How many proofs of the Pythagorean theorem do you know?” Erdős asked.“One,” said Vazsonyi.“I know 37,” said Erdős.(This question was just the second thing Erdős said to Vazsonyi when theyfirst met. The first was “Give me a four digit number,” to which Vazsonyireplied “2,532.” In a second, Erdős came back with, “The square of it is6,411,024.” There was no greeting, no hello, no how are you.∴ Paul Erdős loved math and math alone.)The Pythagorean theorem is one of many theorems with multiple proofs.Multiple proofs are, of course, not necessary for the progress of mathematicsand our desire for certainty. But sometimes finding a totally novel way toprove an established theorem illuminates another aspect of the mathematicalpatterns. The flavor of the theorem can be enhanced by looking at it from anew perspective. Although the Pythagorean theorem has many proofs, manyof them modern, the result of a2 + b2 = c2 maintains the flavor of ancienttimes.Cache Dexter 367Bread is not only one of humanity’s oldest creations, it remains a staple insome form or another in many cultures around the world. Bread wasn’t thefirst food that early humans ate, not by a long shot, but it developed into aconsistent system that has fed billions of humans over the millennia.Besides simply eating to keep ourselves alive, humans feel a need to under-stand and know things. Mathematics may be older than bread itself. Theearliest evidence of counting and number that we have found is the Ishangobone found near the Nile, which clocks in at 20,000 years old. The bone hasa series of tally marks carved into it, and while the meaning of these marksis not clear, the marks indicate that some prehistoric human was attemptingsomething we would describe as mathematics. Bread, on the other hand,didn’t show up until around 14,000 years ago.The study of mathematics has ebbed and flowed across various world cultures,but at the root of it all is the human need to organize and understand.Ancient mathematicians wanted to prove theorems in math, wanted to set instone ideas that they felt were universally true and eternal. They ate breadto fend off starvation, and they proved theorems to fend off uncertainty, toestablish a sense that human activities matter.The ancient Greeks prized idealized geometry, but they couldn’t solve all theirproblems as easily as the Pythagorean theorem. One of these problems wassquaring the circle. The Greeks loved to do geometry with only a compassfor drawing circles and a straightedge for drawing lines, but measurement ofshapes was not allowed. The problem of squaring the circle started with acircle, and the goal was to draw a square with exactly the same area as thecircle using only a compass and straightedge.Figure 1: Taking a circle and drawing a square with the exact same area.The ancient Greek civilization came and went without a solution to squaringthe circle, but not for lack of trying. The Greeks had a word, τετραγωνιζειν,meaning to busy oneself with squaring the circle. They tried mixing a lot ofmathematical dough to bake this theorem. Mathematicians kept attempting368 How to Bake a Theoremto solve the problem for centuries afterward. In this case all their efforts ledto failure, because they were attempting an impossible problem.Theorem. Given a circle in the plane, it is impossible to construct a squarewith the same area as the circle using only straightedge and compass.The Greeks (and everyone after them) had been using the wrong sort of yeastin their pursuits. No amount of mathematical tinkering would ever raise acircle to become a square.Curiously, the proof of this impossibility theorem did not take place in geom-etry but rather in the realm of algebra. Sometimes to bake a theorem, onehas to harvest the flour from a different field of mathematics. Pierre Wantzelproved in 1837, centuries after the Greeks first posed the problem, that inorder to square a circle the number π has to be a solution to a polynomialequation with integer coefficients. There have to be some whole numbersa0, a1, a2, ..., an such that anπn + an−1πn−1 + · · · + a2π2 + a1π + a0 = 0.Finally, in 1882, Ferdinand von Lindemann proved that no such equationexisted, that π was too irrational to be the solution of an algebraic equationlike this.∴ Squaring the circle was officially impossible.Many theorems are conjectured centuries before they are proved, slowly fer-menting and taking on complex flavors. The theorem of the impossibility ofsquaring the circle has another dimension of flavor. Most mathematical the-orems are constructive, describing patterns and processes that you can sinkyour teeth into. To experience the Pythagorean theorem, you can draw a realright triangle. It’s odd to taste a theorem like the impossibility of squaringthe circle, which is fully baked and yet not quite satisfying. Impossibilitytheorems show that even math has boundaries. Perhaps impossibility is anacquired taste.Despite the potential disappointment that squaring a circle is impossible, it iscomforting to know that we can put that particular problem to rest. Knowingthe impossibility is better than not knowing if a mathematical problem canbe solved or not. We can move on to other problems, of which there aremany because math is far from complete. Theorems continue to be bakedtoday, and some conjectures remain open, waiting for someone to come alongand mix together a perfect recipe to finally prove them.Cache Dexter 369The Collatz conjecture sprung up in the 1930s and currently remains un-baked. The conjecture concerns the Collatz process. Start with any positiveinteger. If the integer is odd, multiply it by 3 and add 1. If it is even, divideit by 2. Keep applying these rules to the new numbers you get. For example,if we apply this process to the number 42, we eventually get down to thenumber 1:42÷2−→ 21 ×3,+1−−−→ 64 ÷2−→ 32 ÷2−→ 16 ÷2−→ 8 ÷2−→ 4 ÷2−→ 2 ÷2−→ 1.If we try this with the number 17, then the process also leads to 1:17×3,+1−−−→ 52 ÷2−→ 26 ÷2−→ 13 ×3,+1−−−→ 40 ÷2−→ 20 ÷2−→ 10 ÷2−→ 55×3,+1−−−→ 16 ÷2−→ 8 ÷2−→ 4 ÷2−→ 2 ÷2−→ 1.The Collatz conjecture is that no matter which positive integer we start with,this process will eventually reach 1. Despite how simple the conjecture is, noone has yet been able to prove this.Many people have published various methods they have used to attack thisconjecture. They have drawn trees of the paths that different numbers takeas they collapse down to 1. They have delved into other number systems totry to gain insight on the Collatz process. Nothing has worked yet. It seemshighly likely that the conjecture is in fact true; the process has been verified(by computer) for all numbers up to 100 quadrillion. In every observed case,the Collatz process reaches 1. But no one has yet come up with a real proofthat it works for every positive integer. “Mathematics may not be ready forsuch problems,” said Paul Erdős, who knew 37 proofs of the Pythagoreantheorem at age 17 [1]. Erdős published over 1,500 mathematical papers inhis lifetime; he knew a thing or two about what it takes to bake a theorem.Erdős spent his entire life in the pursuit of mathematical theorems. Henever owned a house of his own, preferring to travel the world, crashing onother mathematicians’ couches in between lectures he gave at conferencesand universities. He couldn’t drive a car, and all of his personal belongingsfit in a suitcase. To him, daily actions such as eating bread were incidentalto being alive enough to do mathematics. The need to prove theorems andbecome certain of things was far greater than the need to do anything else.And yet, Erdős was well aware that there would always be another mathproblem left to solve. He, and every other mathematician, could never staveoff uncertainty permanently.370 How to Bake a TheoremKurt Gödel’s famous Incompleteness Theorem in 1931 forever changed theway humans think about mathematics. Until that time, many people felt thatmath was deterministic, that for any conjecture there was always a proof ofit out there somewhere. Gödel’s theorem shocked the world. His theoremwas a meta-math theorem, a theorem about the nature of mathematics itself.He proved, with completely correct logic, that every consistent mathematicalsystem with the power to do basic arithmetic was incomplete. “Incomplete”means that there are mathematical conjectures that can neither be provednor disproved using the rules of their mathematical systems.Theorem. In mathematics, there will always be statements whose truthful-ness can never be determined.Impossibility may be an acquired taste, but even to those accustomed to itGödel’s Incompleteness Theorem is strong. Gödel baked the ultimate impos-sibility theorem. There are conjectures that can never be proved or disproved,and here’s the kicker: It’s also impossible to know what conjectures are likethis. If someone hands you an unproven mathematical statement, you can’ttell whether it is possible to prove it or not.It may seem like Gödel brought only bad news to the mathematical commu-nity, but there is one upside to his theorem. Since mathematics cannot becomplete, we can rest assured that there will always be some sort of mathproblem to solve. If we come across a mathematical conjecture that we can’tprove, we can decide if we want it to be true or not and use it as an ax-iom, an assumption, instead. This will open the door to another system ofmathematics that will have new conjectures to prove.So we go forward, eating bread today even though we know it is only atemporary solution to hunger, that we will have to bake and eat some moretomorrow. Gödel showed that our pursuit of proofs of mathematical theoremsis no different. We don’t know what theorems we will be able to prove andwhich ones we won’t, but we keep trying, even knowing it is a temporarysolution to our hunger for certainty, that even if we prove a theorem todaythere will be more theorems to prove tomorrow.∴ There is no “ ∴ ” for the whole of math.We will be baking theorems forever.Cache Dexter 371References[1] Paul Hoffman, The Man Who Loved Only Numbers, Fourth Estate Lim-ited, London, 1998.

How to Bake a Theorem

Keck Graduate Institute

Journal of Humanistic Mathematics 
Volume 9 | Issue 2 July 2019 
How to Bake a Theorem 
Cache Dexter 
Follow this and additional works at: https://scholarship.claremont.edu/jhm 
 Part of the Arts and Humanities Commons, and the Mathematics Commons 
Recommended Citation 
Dexter, C. "How to Bake a Theorem," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), 
pages 364-371. DOI: 10.5642/jhummath.201902.31 . Available at: https://scholarship.claremont.edu/jhm/
vol9/iss2/31 
©2019 by the authors. This work is licensed under a Creative Commons License. 
JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and 
published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ 
The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds 
professional ethical guidelines. However the views and opinions expressed in each published manuscript belong 
exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for 
them. See https://scholarship.claremont.edu/jhm/policies.html for more information. 
How to Bake a Theorem
Cache Dexter1
American Fork, Utah, USA
cachedexter@gmail.com
We bake bread to satisfy our hunger, and we prove theorems to satisfy our
hunger for knowledge and existence. Here I explore this analogy.
To bake bread, all you need is flour, water, yeast, and salt. Flour to shape
and form, water to bind and activate, yeast to leaven, and salt to flavor.
To bake a theorem, you also need flour, water, yeast, and salt, of sorts.
A mathematical theorem is a statement about some relationship between
mathematical objects. Importantly, a theorem must have a proof. A proof is
a series of logical steps that trace a theorem back to self-evident mathematical
truths. It isn’t enough to show that a theorem holds for a few or even many
cases. A proof shows that a theorem is true all of the time. A theorem
without a proof is just a guess, a conjecture. It is merely dough, limp and
useless until baked through. It does not satisfy our hunger for certainty.
Theorems are baked with the same ingredients as bread, in a way. Mathe-
matical flour is the raw material, the definitions of the mathematical objects.
A triangle has three straight sides and three angles. If one of the angles of a
triangle measures 90◦, then the triangle is a right triangle. The rules and pro-
cedures of math are the water. These rules activate the definitions in the flour
and form something kneadable and workable. To find the area of a square,
multiply the length of the square by the width. The logic and the flow of the
mathematical argument make up the yeast, the lifting agent. If two squares
have the same side length, then the two squares have equal area. And the
mathematical salt is the details, the wording, the commentary on the math.
1The author teaches secondary mathematics in Utah, USA, and writes about math in
his spare time.
Journal of Humanistic Mathematics Volume 9 Number 2 (July 2019)
Cache Dexter 365
Math isn’t just pushing around symbols; it is the exploration of patterns and
ideas. Pointing out these patterns gives flavor to a proof.
The Pythagorean theorem is an ancient loaf, proved thousands of years ago.
Theorem. Given a right triangle with sides of length a, b, and c, where the
side of length c is opposite the right angle, then a2 + b2 = c2.
Before giving the proof of this theorem, a little aside about proofs. Sometimes
reading a proof isn’t that interesting, or even worth the time. The proof of
the classification of finite simple groups is contained in hundreds of articles
that total over 10,000 pages. In cases like these, only a few people actually
check to make sure the proof is correct and the rest of us just skip to the
statement of the theorem. The statement of the theorem at the end of the
proof is sometimes preceded by a little symbol ∴ , three dots in a triangle
that mean “therefore”. It’s the mathematical symbol for skipping to the end
because the proof was too long, and you didn’t read it.
However, if the proof of a theorem is not very long then reading it may
be valuable. Proofs have a way of showing how mathematics works and
explaining why the patterns are interesting. The proof of the Pythagorean
theorem is not long, but if you feel so inclined you can skip to the “∴” at the
end. Here’s the proof, in a picture:
a
a
a
a
a
aa
a
b
b
b
b
b
b
bb
c
c
c
c
The square on the left contains four right triangles each with sides of lengths
a, b, and c. The remaining area of the white square is c2, since the white
366 How to Bake a Theorem
square has sides of length c. The square on the right is the same size and
contains four identical right triangles, just rearranged. The remaining white
area is formed by two squares, one with sides of length a and the other with
sides of length b. The white area on the right is the sum of these two areas,
which is a2 + b2.
∴ a2 + b2 = c2.
It doesn’t matter what the numbers a, b, and c are. We could draw the same
types of squares starting with any right triangle. That’s what makes this a
proof that the Pythagorean theorem holds for every right triangle. Drawing
a right triangle and measuring its sides are the flour of this theorem. The
rules of geometry are the water, allowing us to draw additional lines and
shapes around the right triangle. The logic of rearranging the shapes without
changing their areas provides the lift, the yeast that finishes the proof that
a2 + b2 does in fact equal c2. A little salt might be that this is only one of
many ways to prove the Pythagorean theorem.
The mathematician Andrew Vazsonyi recounted his first meeting with an-
other mathematician, Paul Erdo˝s, when he was 14 and Erdo˝s was 17 [1].
“How many proofs of the Pythagorean theorem do you know?” Erdo˝s asked.
“One,” said Vazsonyi.
“I know 37,” said Erdo˝s.
(This question was just the second thing Erdo˝s said to Vazsonyi when they
first met. The first was “Give me a four digit number,” to which Vazsonyi
replied “2,532.” In a second, Erdo˝s came back with, “The square of it is
6,411,024.” There was no greeting, no hello, no how are you.
∴ Paul Erdo˝s loved math and math alone.)
The Pythagorean theorem is one of many theorems with multiple proofs.
Multiple proofs are, of course, not necessary for the progress of mathematics
and our desire for certainty. But sometimes finding a totally novel way to
prove an established theorem illuminates another aspect of the mathematical
patterns. The flavor of the theorem can be enhanced by looking at it from a
new perspective. Although the Pythagorean theorem has many proofs, many
of them modern, the result of a2 + b2 = c2 maintains the flavor of ancient
times.
Cache Dexter 367
Bread is not only one of humanity’s oldest creations, it remains a staple in
some form or another in many cultures around the world. Bread wasn’t the
first food that early humans ate, not by a long shot, but it developed into a
consistent system that has fed billions of humans over the millennia.
Besides simply eating to keep ourselves alive, humans feel a need to under-
stand and know things. Mathematics may be older than bread itself. The
earliest evidence of counting and number that we have found is the Ishango
bone found near the Nile, which clocks in at 20,000 years old. The bone has
a series of tally marks carved into it, and while the meaning of these marks
is not clear, the marks indicate that some prehistoric human was attempting
something we would describe as mathematics. Bread, on the other hand,
didn’t show up until around 14,000 years ago.
The study of mathematics has ebbed and flowed across various world cultures,
but at the root of it all is the human need to organize and understand.
Ancient mathematicians wanted to prove theorems in math, wanted to set in
stone ideas that they felt were universally true and eternal. They ate bread
to fend off starvation, and they proved theorems to fend off uncertainty, to
establish a sense that human activities matter.
The ancient Greeks prized idealized geometry, but they couldn’t solve all their
problems as easily as the Pythagorean theorem. One of these problems was
squaring the circle. The Greeks loved to do geometry with only a compass
for drawing circles and a straightedge for drawing lines, but measurement of
shapes was not allowed. The problem of squaring the circle started with a
circle, and the goal was to draw a square with exactly the same area as the
circle using only a compass and straightedge.
Figure 1: Taking a circle and drawing a square with the exact same area.
The ancient Greek civilization came and went without a solution to squaring
the circle, but not for lack of trying. The Greeks had a word, τετραγωνιζειν,
meaning to busy oneself with squaring the circle. They tried mixing a lot of
mathematical dough to bake this theorem. Mathematicians kept attempting
368 How to Bake a Theorem
to solve the problem for centuries afterward. In this case all their efforts led
to failure, because they were attempting an impossible problem.
Theorem. Given a circle in the plane, it is impossible to construct a square
with the same area as the circle using only straightedge and compass.
The Greeks (and everyone after them) had been using the wrong sort of yeast
in their pursuits. No amount of mathematical tinkering would ever raise a
circle to become a square.
Curiously, the proof of this impossibility theorem did not take place in geom-
etry but rather in the realm of algebra. Sometimes to bake a theorem, one
has to harvest the flour from a different field of mathematics. Pierre Wantzel
proved in 1837, centuries after the Greeks first posed the problem, that in
order to square a circle the number pi has to be a solution to a polynomial
equation with integer coefficients. There have to be some whole numbers
a0, a1, a2, ..., an such that anpi
n + an−1pin−1 + · · · + a2pi2 + a1pi + a0 = 0.
Finally, in 1882, Ferdinand von Lindemann proved that no such equation
existed, that pi was too irrational to be the solution of an algebraic equation
like this.
∴ Squaring the circle was officially impossible.
Many theorems are conjectured centuries before they are proved, slowly fer-
menting and taking on complex flavors. The theorem of the impossibility of
squaring the circle has another dimension of flavor. Most mathematical the-
orems are constructive, describing patterns and processes that you can sink
your teeth into. To experience the Pythagorean theorem, you can draw a real
right triangle. It’s odd to taste a theorem like the impossibility of squaring
the circle, which is fully baked and yet not quite satisfying. Impossibility
theorems show that even math has boundaries. Perhaps impossibility is an
acquired taste.
Despite the potential disappointment that squaring a circle is impossible, it is
comforting to know that we can put that particular problem to rest. Knowing
the impossibility is better than not knowing if a mathematical problem can
be solved or not. We can move on to other problems, of which there are
many because math is far from complete. Theorems continue to be baked
today, and some conjectures remain open, waiting for someone to come along
and mix together a perfect recipe to finally prove them.
Cache Dexter 369
The Collatz conjecture sprung up in the 1930s and currently remains un-
baked. The conjecture concerns the Collatz process. Start with any positive
integer. If the integer is odd, multiply it by 3 and add 1. If it is even, divide
it by 2. Keep applying these rules to the new numbers you get. For example,
if we apply this process to the number 42, we eventually get down to the
number 1:
42
÷2−→ 21 ×3,+1−−−→ 64 ÷2−→ 32 ÷2−→ 16 ÷2−→ 8 ÷2−→ 4 ÷2−→ 2 ÷2−→ 1.
If we try this with the number 17, then the process also leads to 1:
17
×3,+1−−−→ 52 ÷2−→ 26 ÷2−→ 13 ×3,+1−−−→ 40 ÷2−→ 20 ÷2−→ 10 ÷2−→ 5
5
×3,+1−−−→ 16 ÷2−→ 8 ÷2−→ 4 ÷2−→ 2 ÷2−→ 1.
The Collatz conjecture is that no matter which positive integer we start with,
this process will eventually reach 1. Despite how simple the conjecture is, no
one has yet been able to prove this.
Many people have published various methods they have used to attack this
conjecture. They have drawn trees of the paths that different numbers take
as they collapse down to 1. They have delved into other number systems to
try to gain insight on the Collatz process. Nothing has worked yet. It seems
highly likely that the conjecture is in fact true; the process has been verified
(by computer) for all numbers up to 100 quadrillion. In every observed case,
the Collatz process reaches 1. But no one has yet come up with a real proof
that it works for every positive integer. “Mathematics may not be ready for
such problems,” said Paul Erdo˝s, who knew 37 proofs of the Pythagorean
theorem at age 17 [1]. Erdo˝s published over 1,500 mathematical papers in
his lifetime; he knew a thing or two about what it takes to bake a theorem.
Erdo˝s spent his entire life in the pursuit of mathematical theorems. He
never owned a house of his own, preferring to travel the world, crashing on
other mathematicians’ couches in between lectures he gave at conferences
and universities. He couldn’t drive a car, and all of his personal belongings
fit in a suitcase. To him, daily actions such as eating bread were incidental
to being alive enough to do mathematics. The need to prove theorems and
become certain of things was far greater than the need to do anything else.
And yet, Erdo˝s was well aware that there would always be another math
problem left to solve. He, and every other mathematician, could never stave
off uncertainty permanently.
370 How to Bake a Theorem
Kurt Go¨del’s famous Incompleteness Theorem in 1931 forever changed the
way humans think about mathematics. Until that time, many people felt that
math was deterministic, that for any conjecture there was always a proof of
it out there somewhere. Go¨del’s theorem shocked the world. His theorem
was a meta-math theorem, a theorem about the nature of mathematics itself.
He proved, with completely correct logic, that every consistent mathematical
system with the power to do basic arithmetic was incomplete. “Incomplete”
means that there are mathematical conjectures that can neither be proved
nor disproved using the rules of their mathematical systems.
Theorem. In mathematics, there will always be statements whose truthful-
ness can never be determined.
Impossibility may be an acquired taste, but even to those accustomed to it
Go¨del’s Incompleteness Theorem is strong. Go¨del baked the ultimate impos-
sibility theorem. There are conjectures that can never be proved or disproved,
and here’s the kicker: It’s also impossible to know what conjectures are like
this. If someone hands you an unproven mathematical statement, you can’t
tell whether it is possible to prove it or not.
It may seem like Go¨del brought only bad news to the mathematical commu-
nity, but there is one upside to his theorem. Since mathematics cannot be
complete, we can rest assured that there will always be some sort of math
problem to solve. If we come across a mathematical conjecture that we can’t
prove, we can decide if we want it to be true or not and use it as an ax-
iom, an assumption, instead. This will open the door to another system of
mathematics that will have new conjectures to prove.
So we go forward, eating bread today even though we know it is only a
temporary solution to hunger, that we will have to bake and eat some more
tomorrow. Go¨del showed that our pursuit of proofs of mathematical theorems
is no different. We don’t know what theorems we will be able to prove and
which ones we won’t, but we keep trying, even knowing it is a temporary
solution to our hunger for certainty, that even if we prove a theorem today
there will be more theorems to prove tomorrow.
∴ There is no “ ∴ ” for the whole of math.
We will be baking theorems forever.
Cache Dexter 371
References
[1] Paul Hoffman, The Man Who Loved Only Numbers, Fourth Estate Lim-
ited, London, 1998.


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