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Extended Essay in

Philosophy

Points: 35/36Grade: A

BE YOUR VERY BEST

Note: This extended essay serves as an example essay that is meant to inspire you in you work with your own extended essay. We hope that it is used for identifying elements that are good to include in order to obtain the grade you strive for. Any plagarism is strictly forbidden.

Is Platonism an adequate explanation

for the nature of π?

IB Extended Essay

Name: Candidate Number:

Subject: Philosophy Session: May 2010 Word Count: 3984

Date: January 19, 2010

Evaluation of Platonism as an Adequate Theory for the Nature of π

2

Abstract

We rely on Mathematics every day, and yet we do not understand what it is. In this

essay I address a branch of this issue by evaluating the adequacy of the Platonic explanation to

the nature of π. I first outline the theory of Mathematical Platonism and briefly introduce π.

Then, I examine some arguments supporting Platonism: the versatility of π, intuition, and two

logical arguments. I then proceed to criticize Platonism, by first refuting the two logical

arguments previously proposed, and then considering other arguments against Platonism, such

as Occam’s Razor, controversy over Dualism, and problems with connecting with the Platonic

world. Finally, I follow Sherlock Holmes’ advice to “look for a possible alternative, and

provide against it”,1 and proceed to consider the two main possible alternatives: Inventionism

and the notion that Mathematics is discovered in the world. I criticize these too. Although the

criticisms are not enough to dismiss the theories for application to all of Mathematics,

therefore leaving that question unresolved, they are enough to establish that both alternative

theories are impossible explanations for the nature of π. Therefore, I ultimately conclude that

although Platonism has several flaws and requires a certain amount of faith, π can only exist in

the Platonic realm.

Word count: 206

1 Sherlock Holmes, Wikipedia, 2010


3

Acknowledgements

I would like to thank my supervisor for his patience, my mother for her impatience,

and my philosophy teacher for inspiring me.


4

Table of Contents

Abstract................................................................................................................................2

Acknowledgements .............................................................................................................3 1. Introduction .....................................................................................................................5

1.1 Mathematical Platonism ............................................................................................5 1.2 π .................................................................................................................................7

2. Advantages of Platonism.................................................................................................7 2.1 π in the physical world ..............................................................................................8 2.2 Platonism is intuitive .................................................................................................8 2.3 The ‘One Over Many’ Argument ..............................................................................9 2.4 The ‘Singular Term’ Argument.................................................................................9

3. Problems with Platonism...............................................................................................10 3.1 Problems with the ‘One Over Many’ Argument .....................................................10 3.2 Problems with the ‘Singular Term’ Argument........................................................11 3.3 Occam’s razor..........................................................................................................11 3.4 Where is the Platonic world? Dualism is implied ...................................................12 3.5 Connecting with the Platonic world ........................................................................12

4. Evaluation......................................................................................................................14 4.1 Discussion................................................................................................................14 4.2 π must be discovered ...............................................................................................14 4.3 The existence of another world is necessary for π ..................................................15

5. Conclusion.....................................................................................................................16


5

1. Introduction

Mathematician and physicist, John D. Barrow said: “by translating the actual into the

numerical we have found the secret to the structure and workings of the Universe”.2

Mathematics seems to explain our world perfectly, and yet why this is so remains a mystery.

Barrow is concerned about the vital role it plays in science. He argues that there seems to be

an underlying belief in our modern society that science will lead us to a Theory of Everything,

a single theory that completely explains our Universe. He insists that since science relies on

Mathematics, any possible Theory of Everything must be mathematical. I believe that this

search for knowledge and understanding of our world is the most beautiful of all human

endeavors, and therefore that it is vital to understand “the meaning and possible limits”3 of the

Mathematical system that we so blindly rely on, so as to avoid running “the risk of building

our house upon sand”.4 If we cannot understand what Mathematics is and why it works, “our

scientific explanations of the Universe are based ultimately upon things we do not

understand”.5 For this reason, I propose to evaluate one of the most popular theories on the

Philosophy of Mathematics: Platonism. In order to keep the discussion within the world limit,

I shall focus on the nature of only one number: π. I shall start by introducing both the theory

and the number under consideration.

1.1 Mathematical Platonism

Number started out as a property6; something immaterial that had to be connected to the

objects it was describing. Mathematics as we know it today began only when number could be

detached from whatever it was describing and considered as an entity in itself. This was first

done by the Ancient Greeks, who had a “highly developed notion of number in the abstract”,7

and since then has shaped our view of Mathematics. In essence, this ancient culture had

2 Barrow, 1993, pvii 3 Barrow, 1993, pviii 4 ibid 5 Barrow, 1993, p2 6 Barrow, 1993 7 Barrow, 1993, p251


6

recognized that in all groups of, for example, four, there was a property of ‘fourness’ that

could be considered separately from any particular group, and most importantly, that any

conclusion about this abstract entity of ‘fourness’ could then re-apply to all groups of four. It’s this element of abstraction that makes Mathematics such a useful tool8: since they

are dealing with pure concepts, mathematicians can apply logic to discover or create other

logically consistent concepts, rather than having to rely on empirical evidence and falsification

for verification. As Einstein said, "as far as the laws of mathematics refer to reality, they are

not certain; and as far as they are certain, they do not refer to reality".9

However, by detaching these ideas from reality, several philosophical issues arise. Do

these abstract entities actually exist, or are they simply the result of human imagination? Is

Mathematics discovered or invented? If the former, a position in epistemology usually referred

to as realism10, then do we derive it from the world or is our world merely a reflection of this

system? Mathematical Platonism, a term first coined by Bernays11, is the theory that

Mathematics exists independently of “the cognitive subject”12, but not in our world:

essentially, that “abstract objects exist”.13 It is “logically equivalent to… realism”14 but it “also

has what might be called a metaphysical component, since it asserts a special kind of

existence, in the space-time continuum, that the objects of mathematical knowledge are

supposed to have”.15

Plato was concerned with the relationship between “particular things and universals

concepts”,16 and he concluded that somewhere ‘out there’, universals must actually exist: the

universal ‘tableness’ that every particular table in the real world shares, exists ‘out there’ as a

concept. He believed that Mathematics deals with universal concepts, and we seem capable to

conceive even those that we can experience only imperfectly, such as perfect lines and perfect

circles. A perfect line, for example, has no width; It therefore cannot exist in the world, and

we cannot even imagine it. However, we can have a concept of it, since we can be considering 8 Mathematics, Wikipedia, 2010 9 Einstein, 1921 10 Mathematical Platonism, PlanetMath, 2008 11 ibid 12 ibid 13 Balaguer, 2009 14 Mathematical Platonism, PlanetMath, 2008 15 ibid 16 Barrow, 1993, p252


7

it now. According to Platonism, this is because Mathematics exists in the Platonic realm: an

eternal and abstract world of ideas. Our world is merely a reflection of this perfect

mathematical universe, and the latter dictates the laws of the former.17 We learn about

Mathematics by exploring this realm through our reason. This leads to the view that

Mathematics is the key to understanding our world, and therefore that our reason alone can

lead us to unlocking the secrets of the universe.

1.2 π

π is “the ratio of the circumference to the diameter of a circle”18. It is intriguing because

it is both irrational and transcendental, which means that it can be expressed neither as a ratio

of two integers, nor as the root of a finite equation.19 It’s ubiquity, which shall be examined

later, is also of importance. I shall now investigate how satisfactory an explanation Platonism

is to the nature of π.

2. Advantages of Platonism

At first sight, Platonism seems to suit the nature of π very well. If we look at a circle

with a radius r, where r is a rational number, then since the circumference, c, is equal to 2πr,

and π is irrational, c will be an irrational number. Therefore, a perfect circle, which should

have a width-less circumference, cannot exist in our world, and as nothing more than a

concept in our minds. To some extent, nothing in the physical world is analogue: everything is

digital, even if only at a level of electrons. Any circle we draw can be only an approximation,

just like we can only ever know an approximation of π. Just like the perfect line, according to

Plato, the perfect circle, and therefore π, must exist only in this eternal world of ideas. This

seems to apply quite beautifully to π, as its irrationality and transcendence alone seem to

suggest that π is a perfect entity beyond our world and understanding. I shall now be

proposing 4 arguments supporting Platonism: π’s versatility, intuition, and two logical

arguments.

17 Barrow, 1993 18 Miller, 2009 19 Pi, Wikipedia, 2010


8

2.1 π in the physical world

The versatility of π has intrigued for years. For example, π’s relevance in river sinuosity

seems quite astonishing.20 π continues to appear in several fundamental laws of physics21 and

in calculus22. However, these connections can all in some way be justified through

connections to the circle. Yet π is also relevant in probability23, and this is most shocking. π is

said to be “ubiquitous in mathematics, appearing even in places that lack an obvious

connection to the circles of Euclidean geometry”24. It “is known for turning up in all sorts of

scientific equations, including those describing the DNA double helix, a rainbow, ripples

spreading from where a raindrop fell into water, waves, navigation and more”25. As Theoni

Pappas said, “it’s startling to discover the versatility of pi, crossing as it does the wide

spectrum of geometry, calculus and probability”.26 What’s intriguing about this constant

unexpected appearance is the philosophical implications on the nature of Mathematics: the

surprising and unexplainable relevance of such a number in seemingly irrelevant areas of

Mathematics suggests that Mathematics is an intricate system that we are still discovering, and

thus that it must exist separate from us. It therefore seems that Platonism is an acceptable

explanation for the nature of π.

2.2 Platonism is intuitive

In fact, Platonism is the most popular theory for Mathematics.27 Hersh and Davis said

that when the typical mathematician is actually doing Mathematics, “he is convinced that he is 20 Henrik Stolum discovered that the ratio between a river’s actual length, from the source to the mouth, and its sinuosity, approaches π. (List of formulas involving π, Wikipedia, 2009) 21 E.g. the cosmological constant, Heisenberg’s uncertainty principle, Einstein’s field equation of general relativity, the equation for magnetic permeability of free space, Kepler’s third law and Coloumb’s law. (ibid) 22 E.g. Euler’s identity and the Gaussian integral. (ibid) 23 E.g. Count Buffon’s needle experiment: If a needle of length d is dropped onto a plane ruled with parallel lines of a distance d apart, then the probability of the needle landing on a line is

€

2π

. π also appears in the Cauchy distribution and the Monte Carlo method, and the probability

of two numbers written at random being relatively prime is

€

6π 2

. (ibid) 24 Pi, Wikipedia, 2010 25 Japanese breaks pi memory record, BBC, 2005 26 Pappas, 1989, p19 27 Barrow, 1993


9

dealing with an objective reality whose properties he is attempting to determine. But then,

when challenged to give a philosophical account of this reality, he finds it easiest to pretend

that he does not believe in it after all”.28 The mathematicians Dieudonne,29 Cohen30 and

Kreisel31 are also quoted to have said something similar. Platonism seems to be intuitive. It

feels right. Even our language reflects Platonism: there seems to be an underlying assumption

in our language that we discover π, and likewise Mathematics, rather than invent it. We speak

of ‘finding’ or ‘discovering’ digits of π, just as we do for solutions, algorithms and theories.

This supports Platonism in that Mathematics is eternal and exists regardless of

mathematicians. Platonists claim that the fifty-billionth digit of π was 2 before we calculated it

to be 2, and this fits with the general idea of the nature of π, as it is seen as a relationship that

we are measuring. In his essay on Mathematical Platonism and its Opposites, Barry Mazur

argues that our experience of Mathematics is the most important thing we have to rely on our

understanding of it, and there is no doubt that “one feels that mathematical ideas can be hunted

down”.32

2.3 The ‘One Over Many’ Argument

The ‘One Over Many’ argument attempts to prove the existence of properties by

proposing a case of several objects that share a property, and as they have this in common,

concluding that the property must exist. For example, if several objects are all, red, these

objects share something and “What they have in common is clearly a property, namely,

redness; therefore, redness exists.”33 To relate the argument to Mathematics, we could simply

change the word ‘red’ to any number, as numbers are also properties, thus asserting that the

abstract concept of numbers exists.

2.4 The ‘Singular Term’ Argument

The ‘Singular Term’ argument was first proposed by Frege in 1884, and goes as follows:

28 Hersh and Davis, 1981, p321 29 ibid 30 ibid 31 Mathematical Platonism, PlanetMath, 2008 32 Mazur, 2008 33 Balaguer, 2009


10

1. If a simple sentence34 is literally true, then the objects

that its singular terms represent exist.

2. There are simple sentences that are literally true and

contain singular terms that refer to abstract objects.

3. Abstract objects exist.35

Parts (1) and (2) of the argument are premises, and if they are true, part (3) follows as a

logically necessary conclusion.

3. Problems with Platonism

Although Platonism is the most popular theory on the nature of Mathematics, it is not

universally accepted. Barrow claims that its element of mysticism makes it much like a

religion: “change ‘Mathematics’, to ‘God’, and little else seems to change”.36 I shall proceed

to consider the most common arguments against Platonism, starting with the refusals of the

proposed arguments for Platonism.

3.1 Problems with the ‘One Over Many’ Argument

The argument proposed in section 2.3 has been considered to be very weak,37 since it

proves the existence of properties, “but not…that properties are abstract objects”38. For

although ‘redness’ has been proven to exist, no reason has been given to believe that it can

exist separately to any object. This relates very well to π. π exists as the property that all

circles in the world seem to share, but this does not mean that it is necessary for π to exist as

an abstract concept. Since the first π (the property) can only be an approximation, as

demonstrated at the beginning of section 2, the implication is that the irrational and infinite π

(the concept) does not have to exist, and therefore that the ‘One Over Many’ argument is not

valid for such cases.

34 “i.e., a sentence of the form ‘a is F’, or ‘a is R-related to b’” (Balaguer, 2009) 35 Balaguer, 2009 36 Barrow, 1993, p296 37 Balaguer, 2009 38 ibid


11

3.2 Problems with the ‘Singular Term’ Argument

The main problem with the argument in section 2.4 is premise 2, as the Platonist would

have to provide examples of such cases “that cannot be paraphrased”.39 The Anti-Platonist

could argue that the sentences such as ‘a is b’ that could be paraphrased to ‘a has property b’,

should be considered as such. This would mean that the abstract objects, or the properties,

would not actually have to exist, as they would be excluded from the argument by the 1st

premise, making the 2nd premise false. This counter argument also applies to π in particular, as

it highlights the previously established fact that π is a relationship between two singular terms,

namely the diameter and the circumference. Therefore, π could not be the singular term of a

simple sentence, as a sentence referring to it would be a variation of ‘d is π-related to c’ with

singular terms d and c.

3.3 Occam’s razor

One possible problem with the theory of Platonism is that the concept of another world

is unnecessary. Occam’s razor states that “the explanation of any phenomenon should make as

few assumptions as possible”,40 and therefore that the simplest explanation is usually the best.

Based on this, Platonism doesn’t seem to be the best theory on the nature of Mathematics

because of its use of an unnecessary assumption. Plato speaks of an abstract world. But is this

world actually necessary? Philosophy of religion could be dismissed with the same argument:

believing in a God who is an invisible being that I will never know actually exists throughout

my life is the same as not believing in him.41 His existence has the same impact as his

inexistence, and since his inexistence makes the least assumptions it is more rational to accept

this as a conclusion.42 However, there are also arguments against Occam’s razor. For

example, Einstein said that “everything should be made as simple as possible, but not

simpler”.43 Looking for the simplest explanation could lead to over-simplification if we are not

aware of all the facts, which could often be the case.

39 ibid 40 Occam’s razor, Wikipedia, 2010 41 Dawkins, 2006 42 This argument is proposed by Antony Flew in his “Parable of the Gardener” (SJG Archive, 2010) 43 Shapiro, 2006, p231


12

3.4 Where is the Platonic world? Dualism is implied

As some philosophers argue, Platonism relies on a “frustrating vagueness”.44 If we do

accept this concept of another world, then the question arises of where this world actually is.

For the sake of this essay, in order to avoid the controversy over the Epistemological

Argument,45 we shall assume that the Platonic realm exists within space and time. Since the

platonic realm is a world of ideas, this does not mean that we have to be able to locate it with

our senses. This does weaken the Platonist’s argument, however, as it uncovers several

assumptions at its foundation. If this world exists separate from us and we access it with

something that isn’t our senses, then something beyond our senses must exist. We therefore

rely on Descartes distinction between mind and body,46 as we must have a soul, or a mind, that

is separate to our body and can access a world that all other minds can access too. If we do not

have bodies, and only minds, then the world is all a figment of our imagination and therefore

there cannot be any value, truth or meaning to Mathematics. If, instead, we are only body and

no mind, then the platonic realm cannot exist as it is a world of ideas and we would have no

soul or mind to explore the realm with. Dualism is a very controversial theory,47 and yet

Platonism can only work if it is true.

3.5 Connecting with the Platonic world

Even if we did assume dualism, Platonism leads to another question: how do we

connect with this world? Since this world is abstract, we must connect to it with our reason:

our mind, or as Plato put it, our soul.48 An alternative is presented by the authors of The

Mathematical Experience, with regards to Platonism: “the job of the theorist is to listen to the

universe sing and write the world down”49 as Mathematics expresses itself through our

universe. However, this suggests that Mathematics is done by experiment, which is the

approach to science, whereas mathematical theorems are arrived at through logic. An example

44 Barrow, 1993, p273 45 According to the Epistemological Argument, human beings cannot attain knowledge of abstract objects that exist outside of the spacetime continuum. (Balaguer, 2009) 46 Descartes, 1986 47 There are several arguments against Dualism (Dualism (philosophy of mind), Wikipedia, 2010) 48 Platonism, Wikipedia, 2009 49 Hersh and Davis, 1981, p69


13

of how Mathematics cannot be done by experiment is the controversy over the 4-colour

theorem, which was proved by trying all the possible outcomes on a computer, and is therefore

not actually considered to be a theorem by most Mathematicians.50 Barrow argues that the

computer had “contaminated the domain of pure unaided thought that they believed

mathematics to be”.51 He then proceeds to highlight that what was most disappointing was not

the use of the computer itself, but the examination of all the cases.52 He even claims that rather

than proving new results, what most mathematicians actually do is “find simpler, shorter and

clearer ways of proving results that are already known”.53

Therefore, it seems that, if Platonism is the case, we discover Mathematics by

connecting with the Platonic realm. But is this a skill that we have and can develop or just a

special ability that only some people have? Mathematics is taught, and this suggests the first,

and yet some people are better at Mathematics than others, which seems to deny the first and

suggest the second. This contradiction undermines Platonism. Geometry, however, is

different. Euclidean geometry is not difficult to apprehend, due to its close connection to the

world, and not a lot of people find it too difficult.54 Plato believed that our knowledge of

geometry ‘is not a memory or the result of education: it is an example of our ability to

perceive the contents of a changeless world of universal truths”.55 However, since when we

are solving an algebraic problem, or even a geometrical one, we don’t feel as if we are actually

visiting another world, Platonism is counterintuitive. We just feel as if we are using our logic

to apply what we have learnt to the problem in order to solve it. In fact, the quote above could

be reworded to say ‘…our ability to perceive the changeless aspects, (and therefore, perhaps,

universal truths) of our world’ to deny Platonism, remain intuitive, and eliminate the

seemingly unnecessary assumption of another world.

50 Barrow, 1993 51 Barrow, 1993, p230 52 ibid 53 Barrow, 1993, p233 54 In his Dialogues, Plato illustrates an episode in which Socrates shows that an un-educated slave-boy has some knowledge of geometry (Gottlieb, 1997) 55 Barrow, 1993, p253


14

4. Evaluation

4.1 Discussion

Having refuted the formal arguments proposed in sections 2.3 and 2.4 in sections 3.1

and 3.2, respectively, and having shown, in section 3.5, that Platonism is also, in some

respects, counterintuitive, it seems that the strongest argument for Platonism as an adequate

theory for π is π’s relevance in the world. However, there are two problems with this. The first

is that π’s versatility doesn’t actually promote Platonism, but just that Mathematics is

discovered. In fact, Mathematics being discovered in the world, rather than found in an

abstract realm of ideas, can seem a far simpler and more appropriate solution. The second is

that the argument isn’t even very strong; as one could say that if we invent Mathematics then

it is a system that we force to adapt to our world. The connections between seemingly

irrelevant areas of Mathematics, such as π and probability, may seem abstract as we do not

know them all, but just because we do not know them does not mean we did not create them.

An alternative to Platonism that addresses both these problems would have to claim that

Mathematics is invented. Since space precludes a full examination of all alternatives, we shall

consider Inventionism.56 The aim is not to explore Inventionism fully, but to show that any

alternative to the theory that Mathematics is discovered, is not only problematic, like

Platonism, but also impossible when considering π in particular.

4.2 π must be discovered

Inventionism is the theory that reduces Mathematics to “something that comes out of the

mind instead of something that enters it from outside”.57 For example, Galileo said that “if the

ears, the tongue, and the nostrils were taken away, the figures the numbers, and the motions

would indeed remain, but not the odors, nor the tastes, nor the sounds, which without the

living animal, I do not believe are anything else than names”.58 According to the Inventionists,

if the mind were removed, then Mathematics would no longer exists, as it is nothing more than

56 For the purpose of this essay we do not distinguish between Inventionism and other related philosophies (e.g. Constructivism and Formalism), which have similar propositions. 57 Barrow, 1993, p294 58 Barrow, 1993, p150


15

a ‘name’, or a reduction of reality that our brain has carried out in order to simplify the

complex.59 According to Inventionism, there is no perfect π, no ideal circle, no width-less line

and no other world: these are only concepts and only the imperfect Mathematics in our

physical world actually exists. This leaves us just with our ideal Mathematical system: a

compression of reality, but nothing more than a concept.

Barrow refutes Inventionism by arguing that it would be impossible for human beings to

have survived for so long if the mathematical systems that we have relied on were not

“accurate representations of the true nature of the external mathematical reality of the

world”.60 Another argument against Inventionism is the objectivity of Mathematics. The fact

that, for example, Pythagoras’ Theorem61, had also been separately discovered in China,62

weakens Inventionism because it implies that if Mathematics is purely a human creation, all

humans must share a capacity that is completely independent from the physical world, culture

and time period.

Although this shows that an alternative to Platonism could be just as problematic, the

difficulty that forces us to completely abandon Inventionism within this essay is its

implications on the nature of π: π wouldn’t exist. Inventionism does not suit the nature of π

since it reduces π to a human construction and nothing more than a concept with no existence,

denying both π’s nature as defined in section 1.2 and it’s ubiquity as established in section 2.1.

4.3 The existence of another world is necessary for π

What seems to be the most difficult Platonist aspect to accept is the concept of another

world.63 However having shown that π must be discovered, it follows that π must exist in

another world, as proposed at the end of section 3.5. This is due to a distinction that seems to

have emerged throughout this discussion, and particularly in section 3.1: the discrepancy

between π as a property (

€

π P ) and π a concept (

€

πC ).

€

πC is

€

π as defined in section 1.2,

whereas

€

π P is only present in the physical world. However, whereas

€

πC is infinite,

€

π P

59 Barrow, 1993, p164 60 Barrow, 1993, p295 61 Which states that in a triangle with sides of length a and b, and hypothesis of length c,

€

a2 + b2 = c 2 (Pythagorean Theorem, Wikipedia, 2010) 62 Pythagorean Theorem , Wikipedia, 2010 63 As shown in sections 3.1, 3.3, 3.4 and 3.5


16

currently has only 2.7 trillion digits.64 Although this number is always increasing, we know

that

€

π Pwill never be equal to

€

πC . For this reason, it does not suit π’s irrational nature to exist

as

€

π P . Therefore, in order for it to exist, π must exist as

€

πC , such that the abstract concept of

π must exist.

5. Conclusion

Although Platonism suits π in certain aspects, it introduces numerous dilemmas. Once

we actually think beyond the idea of Mathematics that even our language has conditioned us

to believe in, Platonism can actually seem absurd. Barrow has called it “the strangest, and yet

the most familiar of the possible Mathematical worlds”.65 In Platonism and Anti-Platonism in

Mathematics, Balaguer shows that there are no good arguments for or against mathematical

Platonism, and that there never will be.66 This supports the claim that “the Platonists’ belief in

a realm of mathematical objects is like belief in the existence of God because one is at liberty

to disbelieve it if one chooses”.67 But what, then, is the best theory to believe?

Platonism requires an element of faith, and this weakens it significantly as a theory. It

asks for too much and makes too many assumptions. Placing π in the ‘sky’ and claiming that π

in the world is a reflection of this more perfect π seems extreme, when we could just assume

that π is in the world already, rather than in the ‘sky’. In fact, one could refuse both Platonism

and Inventionism, and believe that when Pythagoras claimed that “things themselves are

numbers”68, he was perhaps closer to the truth: that “We are part of the Platonic world not

communicators with it”.69

Which one of the theories proposed best suits Mathematics remains an unsolved

question, yet it seems we can make a conclusion regarding π. If we are considering π in

particular, there is a problem with having π in our world, as established in section 4.3: it

reduces π to just an estimate, or

€

π P . So we are left either with Platonism and the faith it

64 Palmer, 2010 65 Barrow, 1993, p296 66 Balaguer, 1998 67 Barrow, 1993, p297 68 Barrow, 1993, p251 69 Barrow, 1993, p297


17

requires, or with only a system of invented abstract concepts that do not exist. But π only

exists in the former. Therefore, we conclude that, as imperfect as it is, Platonism is the only

explanation for π. π can only exist in the sky.


18

Bibliography

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