Chapter 2:
Reply to Usual Objections to Platonism
2.1
- Causal Knowledge for Legitimate Epistemology?
2.2 -
Semantic Fictionalism
2.3
- Non-Twilight Zone View of Formal Knowledge: Platonist
Epistemological Theories
2.3.1
- The Mind's Eye
2.3.2 - Realistic
Rationalism
2.3.3 -
Husserlian Epistemology of Mathematics
2.4 -
Scientific Evidence for Mathematical Intuition
2.5
- Platonism and the Fallibility of Knowledge
2.6
- Where and Why Do Mistakes Occur?
There are many philosophers who reject mathematical platonism on a
basis very similar to that expressed by Michael Dummett:
If
mathematics is not about some particular realm of empirical reality,
what, then is it about? Some have wished to maintain that it is indeed
a science like any other, or rather, differing from others only in that
its subject-matter is super-empirical realm of abstract entities, to
which we have access by means of an intellectual faculty of intuition
analogous to those sensory faculties by means of which we are aware of
physical realm. Whereas the empiricist view tied mathematics too
closely to certain of its applications, this view generally labeled
“platonists,” separates it too widely from them: it leaves
unintelligible how the denizens of this atemporal, supra-sensible realm
could have any connection with, or bearing upon, conditions in the
temporal, sensible realm that we inhabit.
Like the empiricist view, the platonist one fails to do justice to the
role of proof in mathematics. For presumably, the supra-sensible realm
is as much God's creation as is the sensible one; if so, conditions in
it must be as contingent as in the latter.
59
In other words, here we find three objections to platonism in general.
The first one is of epistemological nature, if abstract objects cannot
be perceived or have any causal relationship with us, how do we know
them? The second has to do with the question about which intellectual,
mental or psychological ability do we use to know them. Finally, are
these abstract objects contingent (God's creations)?
This chapter will address these usual objections to platonism. It will
not be an exhaustive refutation of every argument against platonism
presented in philosophy. However, it will provide enough answers to
cover the necessary bases to explore the extent of the relationship
between formal and natural sciences, and see if
a posteriori
matters-of-fact can revise logic and mathematics.
Before we begin, we must point out that Benacerraf (1983) challenged
philosophers of mathematics in general to formulate a philosophical
doctrine of mathematics that fulfills the following requirements:
- First,
any proposed good philosophy of mathematics should reach the
objectivity and high level of consistency of mathematics, thus
providing an adequate account of mathematical truth. This has been
accomplished by platonism.60
- Second, it should provide an adequate epistemology of
mathematics, which supposedly platonism is not able to offer.61
We
will certainly address the problem presented in the second requirement
that, according to Benacerraf, platonism is not able to satisfy.
2.1
— Causal Knowledge for Legitimate Epistemology?
Platonism as such is not positing objects “floating in the middle of
the air”, or “creatures of God”, or anything similar. We are talking
about an abstract reality that is absolutely necessary, and is the
formal
condition of
possibility of any object or any state-of-affairs whatsoever.
Therefore, we are referring to an ideal realm autonomous from human
psychology and the physical world, and in the specific case of pure
logical and mathematical truths, completely independent. We have to
add, as Husserl pointed out, that we are not talking about numbers or
other ideal objects and concepts as forming part of the real (
reell) world, which
would lead to a “metaphysical hypostatization”.
62
This abstract nature of logical and mathematical entities makes it
impossible to establish a causal relation between such ideal realm on
the one hand, and the psychological acts and physical world on the
other. This goes against certain philosophers' doctrines that ask for
causality as a condition for all acceptable epistemology. The
requirement of causality for an adequate epistemology of mathematics
does not seem convincing for many reasons. First, it is not
self-evident. Our view of mathematical epistemology is non-causal, the
abstract entities themselves do not cause our knowing them, and yet,
our cognitive processes are perfectly natural. Whichever causal
mechanisms our brain or mind uses to constitute objectualities and
essences do not depend causally on numbers or ideal entities themselves.
Second, in natural science, the criterion of causality is not applied
to all cases. For example, in quantum physics we cannot establish
certain causal relations among subatomic particles, and yet we have
legitimate scientific knowledge about them. The following example shows
one case:
Illustration 1
Illustration 1 shows an EPR Machine.
63 Let us suppose
we have a positronium
64
that annihilates and two photons are released in opposite directions.
According to experiments carried out by scientists, one photon will
spin in one direction and the other will spin in the opposite
direction. Apparently, there are no exceptions to this phenomenon. We
could conjecture why this happens, but it seems we cannot formulate a
theory that can causally link both spins. We could say that this
happens because one photon seems to affect the other. Since photons
travel at the speed of light, the effect of one on the other should be
faster than the speed of light, which goes against special relativity's
assertion that nothing can travel faster than this speed. We could
perhaps think that what affects their spin is at the very source of the
photons at the moment of annihilation, but quantum physics states that
if such a thing were true, the outcome would be completely different.
Therefore,
we can know
non-causally the spin of one photon by knowing the spin of
the other one. The causal requirement for legitimate knowledge is
refuted.
65
2.2
— Semantic Fictionalism
Some philosophers of science like Mario Bunge are not completely
satisfied with platonist proposals, not even Popper's semi-platonism.
Bunge chooses fictionalism with respect to propositions, logic, and
mathematics as an alternative. He says that there are no
“propositions-in-themselves” as Bernard Bolzano, Gottlob Frege, Edmund
Husserl and others seem to believe, and says that we have to make
believe that they exist. He says that unlike Mickey Mouse or Superman,
66
we can establish formal and semantic exactness using these fictitious
notions.
67
Since his conception of meanings and propositions is fictionalist, the
truth value for these propositions is linked to human psychology and to
scientists' and mathematicians' “conventional knowledge”. He gives the
following examples to show that not all propositions are true or false,
and that there are other propositions with no truth value: the trivial
propositions “the trillionth decimal character of π is 7”, “in the
center of the Earth there is a piece of iron”, and the non-trivial
proposition “The value of function
f,
representative of a
P
property for an individual
x,
is
y”
where
f
is an attribute of the figure in a scientific theory. Bunge says that
in these and other cases a proposition lacks truth value, not because
one has not been assigned to it, but because it is impossible to decide
if it is true or false.
68
In light of platonism, Bunge's real problem is that he supposes that a
proposition is true or false only when we
know
such truth value on the basis of facts. For platonists, it would be as
absurd as claiming that the proposition “the Sun is in the center of
the Solar System” was not true one time because human beings did not
know it. The proposition just expressed has always been (and is)
objectively true, even when throughout the centuries humanity believed
the false proposition “the Earth is the center of the universe”. That
humanity did believe that “the Earth is the center of the universe” is
true did not make such a proposition true. Bunge confuses holding a
proposition as true with it
being
true.
For a platonist, it is possible for a proposition to have a truth
value, even if we do not know it, or if we do not know the proposition
itself. Let us take the example of “the trillionth decimal character of
π is 7”. This proposition will depend ontologically on the number π
itself. According to platonists, there is a trillionth decimal
character, but we
do
not know
which one it is. Knowing and being are two very different things.
Unfortunately, Bunge's own conception of numbers as being only products
of human imagination leads him to confuse knowledge of the number with
its being, and he imagines that this is enough to refute platonism.
Epistemological limitation is not the same as lack of truth value of a
proposition.
With respect to “the value of function
f, representative
of a
P
property for an individual
x,
is
y”,
this itself is a function, not a proposition. Its truth value will
depend on the values to be assigned to,
P,
x, and
y.
Once the assignment is made, then a proposition is expressed. At least
in logic we have made a very clear difference between a propositional
function such as “
F(
x)” where
x is a variable,
from singular logical statements such as “
F(
a)” (where a is a
constant), or as propositions where a quantifier is prefixed such as:
((∀x)(F(x)→G(x))∧(∀x)(G(x)→H(x)))→(∀x)(F(x)→H(x))
As Bunge clearly stated, truth value can only be applied to
propositions. If this is correct, they are not applied to functions.
Bunge develops an epistemological
doctrine about propositions which does not let us talk about
all
propositions as being true or false. His proposal overcomes some
difficulties with Quine's statements about the problem of identity of
meanings.
69 For
him, it is possible to say that there are propositions, just like
mathematics can state that there are numbers, but without stating that
there are “propositions-in-themselves” or “numbers-in-themselves”.
In light of this epistemological doctrine, he establishes the
unequivocal ontological existence of physical organisms capable of
thinking and forming judgments. Possibly there are
no identical judgments,
nor an identical process in two or more brains, or even in the same
brain. However, there are
similar
judgments.
If that similarity is pointed out enough, then we can conclude that two
brains are thinking the same proposition. This leads him to think that
concepts, propositions, and numbers lack autonomous existence. At most,
we can define their existence psychologically: “
x exists
conceptually if there is at least one brain that thinks
x”.
70
Apparently, Bunge is not aware that
platonists already have replied to such statements. For example, in
The Foundations of Arithmetic,
Gottlob Frege addressed precisely this kind of psychological view of
propositions and numbers. Such view states that there are no identical
propositions or notions, since they are subject to psychological
change, and subjective notions cannot be shared. Let us suppose, for
the sake of the argument, that mathematical objects are figments of our
brains' neurons and nothing else beyond that. If there is no abstract
ideal realm, nor a cultural realm that is essentially abstract, it is
difficult to account for our agreement about propositions and
mathematical entities.
71
Throughout history and in every society, humans have represented
numbers in very different ways (think about Greek numbers, Roman
numbers, Hebrew numbers, Arabic numbers, and so on), and yet, they all
have the same
meaning,
and
refer to exactly the same objects without there being any difference
between the properties of the Arabic “1”, the Hebrew “א” or the Roman
“I”. We can all agree that the number five is a prime number, the
result of adding 2 and 3, the result of adding 1 and 4, the number
whose square is 25, and whose cube is 125, and so on.
72
Bunge does not solve the problem of defining numbers as subjective
representations. Each representation of the number one (as similar as
it can be to other number ones in other people's brains)
will always be different
with different subjective characteristics. So we cannot talk about the
number one. Instead we would be talking about 1, 1', 1'', 1''', and so
on.
73 It would be a miracle if
rational beings can understand one proposition independently of
non-shared mental representations.
2.3
— Non-Twilight Zone View of Formal Knowledge: Platonist Epistemological
Theories
Platonist epistemology is often accused of being mysterious and
mystical. As Katz (1998) has pointed out, the fact that something is
mysterious does not make it illegitimate. Philosophy is replete with
mysteries, and it thrives in solving those mysteries rationally.
74
It has been accused of mysticism, because it would require a kind of an
extra-eye to “see” those mysterious entities floating in the air
waiting for someone to grasp them and know them. Mysticism proposes
that we can obtain knowledge beyond our cognitive faculties.
75
In this section, we will see that this is not so. First, we will see
three platonist theories regarding formal knowledge, choose one of
them, and
then
look at empirical evidence for this in animals and humans.
2.3.1
— The Mind's Eye
Several platonist philosophers of mathematics like James Robert Brown
posit the existence of “the mind's eye” which, in many ways, is
analogous to the physical eye. We must understand the term “mind's eye”
only as a metaphor to describe that aspect of the mind that lets us
access the mathematical realm.
76
For example, says Brown, we ordinarily perceive a cup of coffee on the
table, and we are compelled to believe in the actual existence of the
cup. In the same way, we are compelled to believe in the actual
existence of mathematical objects and truths, such as the necessity of
the proposition “2 + 2 = 4” and the contingency of the judgment “the
bishops move diagonally”.
77
An objection may be raised that we know more or less the physical
processes by which our physical eyes perceive objects. On the other
hand, the way this mind's eye perceives needs to be explained.
78
Brown argues against this objection saying that this mystery of the
“mechanics” of the mind's eye does not undermine this platonist claim
that we perceive abstract mathematical objects. It is a fact. As we
have seen, the criterion of causation has been refuted, so opponents
cannot use it against platonism.
79
Finally, Brown argues in favor of the
concept of the mind's eye by recognizing the fallibility of formal
a priori
knowledge. As Descartes argued very well, perceptions can deceive us.
What we perceive does not necessarily correspond to the actual facts.
What seems to be a lake from afar may be, in reality, a hallucination,
an illusion, or a mirage. The same thing happens to the mind's eye.
When we formulate a mathematical conjecture that is not self-evident,
we may be subject to error. He calls this philosophical standpoint
“fallibilist platonism”.
80
The epistemological proposal of the mind's eye has one advantage, and
it is precisely fallibilist platonism. As I will show later in this
chapter,
a priori
knowledge can be fallible.
However, the “mind's eye” doctrine has several serious disadvantages.
All that this doctrine says is that we perceive abstract objects, but
it does not begin to explain
how
we perceive them. Also, it seems that the analogy with the physical eye
is flawed. In the realm of all possible worlds, we can see as possible
that the world as we experience and perceive it is an illusion, and
that our compelling belief about the existence of a cup that we
perceive may not be true. But in the case of formal knowledge there
seems to be no possible scenario where the principle of
no-contradiction is false, or that the square root of two is not an
irrational number. In this case,
a
priori knowledge seems to be far more compelling than an
a posteriori
knowledge of the existence of a cup based on physical perception.
We also need to recognize another aspect where the analogy with the
physical eye fails. Let us assume, for the sake of the argument, that
it is still a mystery today how our brain works regarding the way
physical perception operates. We can still argue that despite this
mystery, something seems to be
given
to us. Even if the world were an illusion, it would still be reasonable
to suppose its existence. So, even if we don't know how the brain
works, we know that there must be a natural mechanism for perception.
The problem with abstract mathematical objects is that they are not
perceived sensibly, which leads to several hypotheses that would seem
prima facie
viable regarding our mathematical knowledge: numbers are conventions,
or fictions, or arise from sensible experience, and so on. These
explanations do seem plausible in relation to natural processes, and
the hypothesis of the “eye” that “grasps” abstract mathematical objects
is not itself powerful enough to be an explanation.
2.3.2
— Realistic Rationalism
Jerrold Katz is one of the most recent philosophers of mathematics who
tried to meet Benacerraf's challenge, and had tried to account for a
platonist epistemology of mathematics without the need to establish a
certain kind of supernatural faculty that grasps abstract entities. He
calls his philosophical position “realistic rationalism”. “Realism” in
this context means that mathematical truths refer to abstract objects:
entities no located in space and time.
81 “Rationalism” means that
reason alone is the source of all formal
a priori knowledge.
82
Therefore, his epistemological theory can be rightfully called
“rationalist epistemology”.
83
Katz establishes several conditions of
an adequate rationalist epistemology:
- There should be no causal relationship between the
knower and the abstract objects to be known.84
- The
truth of a mathematical proposition depends exclusively on the nature
of the proposition itself. For example, the truth of propositions about
physical objects depends on the facts of the physical world. The
knowledge of mathematical truths depends ultimately on their a priori nature and
their epistemological foundation in reason.85
- That
the truths discovered through reason should be justified according to
the objects they talk about, mathematical entities and their properties.86
Contrary to the objects of the external world, whose nature is
contingent, all abstract mathematical entities have necessary
properties. The fact that the square root of two is an irrational
number, and that two is the only even prime number are examples of
this. Both statements about number two are examples of apodeictic
conclusions at which we arrive using
Reductio ad Absurdum.
87
Realistic rationalism recognizes the
existence of a kind of
mathematical
intuition
which is necessary to find these mathematical truths. This is not a
mystical intuition, but rather one that can be explained naturally. He
uses this example:
It is also
crucial to the notion of intuition in our sense that intuitions are
apprehensions of structure that can reveal the limits of possibility
with respect to the abstract objects having the structure. Intuitions
are of structure, and the structure cannot be certain ways. [ . . .]
Consider the pigeon-hole principle. Even mathematically naive people
immediately see that if
m
things are put into
n
pigeon-holes, then, when
m
is greater than
n,
some hole must contain more than one thing. We can eliminate prior
acquaintance with the proof of the pigeon-hole principle, instantaneous
discovery of the proof, lucky guesses, and so on as “impossibilities.”
The only remaining explanation for the immediate knowledge of the
principle is intuition.
88
In this way, Katz refutes antirealists who accuse realists of not
providing an alternative to this mathematical intuition that they
reject as mystical.
It seems by the context
of this epistemological doctrine that this notion of “mathematical
entities” is conceptualized as structures, not as objects in the
fregean sense.
89
For this reason, we find Katz as closer to Nicolas Bourbaki and Edmund
Husserl than to Frege.
Katz also tells us that he favors a kind of fallibilist platonism,
because mathematical intuition itself is also subject to error. The
revision of logic and mathematics is the result of our cognitive
limitations of mathematics, which are eventually superseded through
reason and careful analysis.
90
Katz concludes his presentation of his rationalist epistemology,
showing that there is no begging a question within reason. For him, to
beg the question within rationalist epistemology would be like falling
into a kind of Cartesian skepticism, which questions even the most
elementary mathematical principles. If this is the game empiricists
wish to play against realism, then we should also remember that they
can fall into a humean skepticism if we ask on what logical basis can
we say that we have one single pencil in our hand if all the flow of
experience never remains the same, and that the fact that we perceive
the pencil does not mean necessarily that it exists or that the pencil
is the source of our sensations. Therefore, if the empiricist is
willing to accept within reason (without falling into radical
skepticism) that what we have in our hand has the traits of a pencil,
and that such phenomenon can mean that the pencil does indeed exist,
the empiricist must grant that through
Reductio ad Absurdum
we should show the mathematical truth that, for example, the number 2
is the only even prime number.
91
Also, he pointed out that not holding up some basic logical principles
that are necessary for any rational reasoning would lead immediately
into paradoxes, such as the
paradox
of revisability. The paradox goes like this:
Since
the constitutive principles are premises of every argument for belief
revision, it is impossible for an argument for belief revision to
revise any of them because revising any one of them saws off the limb
on which the argument rests. Any argument for changing the truth value
of one of the constitutive principles must make a conclusion that
contradicts the premise of the argument, and hence must be an unsound
argument for revising the constitutive principle.
92
The principle of no-contradiction is an example. If all logic were
revisable, the principle of no-contradictions should also be revisable.
If it is revisable, it is because certain “events” seem to contradict
the principle of no-contradiction. Since the principle of
no-contradiction should be supposed as a premise necessary to find a
contradiction, then the reason to question this principle is also
questioned, and there would be no base at all for the revision.
Therefore, we reach the conclusion that the principle of
no-contradiction is revisable and it is not revisable.
93
He states that realistic rationalism meets Benacerraf's
challenge, because it explains how we can grasp mathematical entities
and truths non-causally.
94
Katz's advantage over the mind's eye proposal is significant. He shows
the mechanisms through which we can achieve a logical or mathematical
truth without going into metaphors like the “mind's eye”. He also
embraces “fallibilist platonism” and gives us a glimpse of how we are
able to grasp truths about mathematical entities.
However, there is an aspect of Katz's epistemological doctrine that is
not completely satisfied. For example, he implicitly recognizes that we
are able to perceive some mathematical structures in a given
experience. In fact, much later in his book, as we shall see more
thoroughly with Husserl, he says that we are able to see these
structures along with sensible objects. We are able to recognize these
abstract objects once we “purify” them from the sensible components.
95
However, he does not talk much about the way different kinds of
mathematical objects can be found in experience, he just assumes that
they are there and that we can perceive them rationally. More to the
point, his notion of mathematical intuition is not very clear, since he
seems to confuse the “perception” of structures with the intuition that
lets us find their necessary properties or relationships. As we shall
see, his portrayal of mathematical intuition with the pigeon-hole
example would seem a confusion of categorial intuition and eidetic
intuition in husserlian terms.
Finally, Katz
lacks a theory explaining how from these elementary structures we are
able to reach the level of mathematical complexity we know today
(various theories of numbers such as complex numbers, category theory,
universal algebra, general topology, among others)
I do not wish to finish this section without making a comment on a
significant point made by Katz regarding the revisability paradox. As
we shall see, this paradox is a very important stumbling block to
Quine's assertion that in principle all logical truths can be revised.
2.3.3
– Husserlian Epistemology of Mathematics
To address Husserl's epistemological proposal, we have to realize that
it is a result of his epistemological interest in mathematical
knowledge. After his antipsychological turn, in a section called
“Sensibility and Understanding”, in the sixth investigation of his
philosophical work,
Logical
Investigations,
he explained in full detail his phenomenological approach to
mathematics and addresses the problem of mathematical knowledge. To
understand this, we must look at the way judgments or propositions are
fulfilled by states-of-affairs.
According to
phenomenology, intentionality is an essential property or
consciousness, and it consists in directing ourselves to objects. An
intentional act is an act of thinking, a
cogito in the
cartesian sense. Every
cogito
has a
cogitatum,
every act of thinking has an object that is being thought of.
96
As we have seen in Chapter 1, Husserl says that an objectual act, which
is also an intentional act, constitutes an objectuality, a
state-of-affairs. As we have seen, a state-of-affairs is composed of
sensible objects interconnected by categorial forms. For him,
objectualities can be correlated with judgments that are fulfilled in
them (true judgments or propositions). If we say that “Mary is taller
than John”, such a proposition is fulfilled precisely on the objects
Mary and John,
and the way they are related by categorial forms. If we
say, “John is shorter than Mary”, this judgment refers to another
state-of-affairs, another objectuality. The
situation-of-affairs
(
Sachlage),
the sensible objects that serve as a passive basis for both
states-of-affairs, remains the same. If we have sensible objects
a and
b, the propositions
“
a <
b”
and “
b
>
a”
would both refer to two different states-of-affairs, but they are based
on the same situation-of-affairs.
97
This is the way propositions about the temporal world are fulfilled.
Our words can refer to sensible objects: “house”, “book”, “bike”,
“Aristotle”, and many other concepts. However, we cannot disregard
formal words,
which are words with no sensible correlates, like “is”, “the”, “three”,
“below”, “over”, “under”, “then”, “and”, or “first”. These refer to
categorial forms.
98
Given that they have no sensible correlates, how do we know
about them?
An intentional analysis can reveal how states-of-affairs are
constituted. The origin of all our knowledge begins with two sorts of
intuitions:
sensible
intuition and
categorial
intuition. Sensible intuition is what lets us grasp
sensible objects directly. We can identify two kinds of sensible
intuition:
sensible
perception and
sensible
imagination.
The latter lets us constitute sensible objects in our fantasy or
imagination, while the former lets us grasp sensible objects that are
present “in-the-flesh” so-to-speak. Through sensible perception,
sensible objects are given immediately in a low-level objectual act.
They appear as “external” objects before us.
99
On the other hand, categorial intuition
lets us constitute categorial forms
on the basis of
sensible objects. In this sense, they are higher-level formal concepts
that relate low-level sensible objects.
100
However, these categorial forms are not given in the same manner than
sensible objects, because they require the intervention of
understanding through categorial acts. In the act of constituting a
state-of-affairs through a categorial act, objects are given in a
specific manner. We
can either talk about a
set
of pencils, or about
five
pencils or the
total
of pencils, and other kinds of states-of-affairs based on one
situation-of-affairs.
In all of these cases, we treat sensible
objects as a
unity of
experience.
This lets us use these states-of-affairs as basis for other categorial
or objectual acts, there can still be higher objectual levels.
101 This can be clearly
seen in the case of sets. He gives us the following example:
In
the domain of receptivity there is already an act of plural
contemplation in the act of collectively taking things together; it is
not the mere apprehension of one object after the other but
retaining-in-grasp of the one in the apprehension of the next, and so
forth [. . .]. But this unity of taking-together, of collection, does
not yet have one object: the pair, the collection, more generally, the
set of the two objects. It is limited consciousness, we are turned
toward one object in particular, then toward another in particular, and
nothing beyond this. We can then, while we hold on the apprehension,
again, carry out a new act of taking together [of, let us say] the
inkwell and a noise that we have just heard, or we retain the first two
objects in apprehension and look at a third object as separate from
others. The connection of the first two is not loosened thereby. It is
another thing to the combination or to take a new object into
consideration in addition to the two objects already in special
combination. And then we have a unity of apprehension in the form of
{{A,B},C}: likewise {{A,B},{C,D}}, etc. It is necessary to say again
here that each apprehension of a complex form has as objects A B C . .
. and not, for example {A,B} as one object, and so on.
102
In other words, the example of sets shows us that there can be a
hierarchy of objectualities. If we wish to find the sensible subtracts
of all of these objectual acts, we can trace them down to the sensible
components.
103
Of course, in states-of-affairs we find sensible components and formal
components, and we are able to distinguish matter and form. They are
essentially the result of
mixed
categorial acts
where we constitute a state-of-affairs which include empirical content
and formal content. As we have seen, every categorial form given in
categorial intuition rests on sensible intuition.
104
This has to be distinguished from
pure categorial acts
where
pure categorial
concepts
are constituted without any sort of sensible components. We know they
are constituted, because all of the
mathesis universalis is empty of
all sensible content.
105 How can it be
constituted?
To reach the level of pure categorial
forms, an
act of
categorial abstraction
is needed. With this act of understanding, we purge a state-of-affairs
of all of its sensible contents, and stay with the categorial forms. In
this process, all sensible concepts are substituted with indeterminates
(variables), and the only thing that matters in mathematics and logic
is the essential relationship between these categorial forms.
106
In the case of mathematics we explore the necessary or essential
relationship between formal-ontological categories: unity, plurality,
relation, cardinal numbers, ordinal numbers, part and whole, and so on.
What lets us constitute these essences (necessities or possibilities)
is what Husserl called “
general
intuition”, “
essential
intuition” or “
eidetic
intuition”.
Through eidetic intuition, essences are constituted, and we are able to
discover the necessary relationships between concepts, be them material
or formal.
107
Also, Husserl explains that to understand logic, we must understand
that the acts by which we constitute a state-of-affairs, while we are
able to formulate judgments or propositions through meaning acts. The
most basic of these meaning acts is expressed by the concept of
“being”. Once we constitute a state-of- affairs, through meaning acts
we establish that such a state-of-affairs is the case. If we find Mary
taller than John as a state-of-affairs, through a meaning act we can
express that “Mary
is
taller
than John”. “Is”, in this case, is a meaning category constituted by a
meaning act. Through more meaning acts we can formulate judgments such
as: “Mary
is
taller than John
and
John
is
shorter than Mary” where we establish a conjunction among elementary
propositions. We can abstract a pure logical truth once the material
concepts in propositions are substituted by indeterminates or variables
(categorial abstraction). We can obtain from such abstraction true
subject-predicate forms, disjunction, conjunction, and so forth: “A
is B”, “A
and B”, “A
or B”, and so on.
108
Since every objectual act can have a meaning act as correlate, and
given the fact that there are unlimited ways to form complex
propositions about objectualities, we can also form a hierarchy of
meanings which can be traced down and be founded on sensible
components, and their complication also has to follow laws, namely, the
“laws to avoid non-sense”, which are the laws of pure universal
grammar. The essential relationship between meaning categories and
between any sort of propositions can be discovered through categorial
abstraction, as we explained above. Analytic laws are discovered when
these judgments, purified of all material concepts, follow the laws to
avoid non-sense, and the laws to avoid counter-sense in order to
preserve truth.
109
For Husserl, we are able to reach the
mathesis universalis from these
pure categorial acts. From a transcendental standpoint we can
distinguish between what he called “
formal
logic” and “
transcendental
logic”.
For Husserl, the world is already given in a determined and
determinable way to transcendental consciousness. It is to this
experience that we must turn to, so we can understand the relationship
between the subjective forms of reasoning (transcendental logic) and
the objective logic (formal logic) in its supreme form as
mathesis universalis.
As we will show, he argued that transcendental logic turns to objective
formal logic, while formal logic needs transcendental logic to be
constituted.
110
As we have seen in
Chapter 1, logic consists in three different strata.
From a transcendental point of view, each one of these strata is
further removed from psychology until it constitutes formal logic. The
first stratum on the side of logic is the morphology of meanings or the
universal pure grammar, which follows the laws to prevent non-sense. On
the side of mathematics we find the
morphology of intuitions
or
morphology of
formal-ontological categories.
111
Logic's second stratum builds on the first one, and deals with forms of
deductions and demonstrations, and its laws, called by Husserl “laws to
prevent counter-sense”. This stratum is still syntactic. However, he
makes a distinction between the “logic of consequence” and the “logic
of truth”. The latter includes the concept of truth among other related
concepts. In other words, there must be a difference between syntax and
semantics. On the side of mathematics we also find a second stratum
that deals with theories based on formal-ontological categories
completely purified from any empirical content, such as set theory,
theory of plurality, theory of numbers (arithmetic), and so on. It
simply consists in theories of possible objects and states-of-affairs
based formal-ontological categories, in other words, based on
mathematics' first stratum.
112
Finally there is the third stratum, where logic becomes a theory of
deductive systems or a theory of all possible forms of theories. On the
side of mathematics, its supreme form is a theory of manifolds. Both of
these correlated strata become a
mathesis
universalis
where we only reason completely on a level of pure forms, and purified
from all empirical content, whose truth cannot be reduced to
psychological activities. This
mathesis universalis, from an epistemological standpoint,
is the
a priori
basis of any and every science in the broadest sense.
113
Here we illustrate a summary of Husserl's doctrine regarding these
three strata of formal logic and how we reach this
mathesis universalis.
Illustration 2
The advantage of Husserl's epistemology
over the other two are great. First, it can tell us
how
from our experience of states-of-affairs we are able to abstract
mathematical objects, or formal-ontological categories. We constitute
them through objectual acts, purify them from all empirical content,
and then reach the theory of manifolds in the husserlian sense. It does
not state just that there is a “mind's eye”, but Husserl explains that
due to the fact that we constitute states-of-affairs through mixed
categorial acts, we can perceive sensibly the objects, but
we can also perceive the formal
relationship between them.
Katz is close to this notion when he talks about mathematical
relationships that need to be purified of all empirical concepts or
objects.
However, there is a difference
between what Katz calls “mathematical intuition” and what Husserl
conceives as mathematical intuition. For Katz, mathematical intuition
is closer to a mix of categorial intuition and eidetic intuition, with
which we are able to “perceive” the
necessary and
possible
relationships among formal structures. On the other hand, for Husserl,
mathematical intuition is a pure categorial act: categorial intuition
and categorial abstraction. That is, we constitute a state-of-affairs
and then, through an act of formalization, we obtain the mathematical
objects themselves purified from all empirical content. Once these
formal-ontological categories are constituted in their pure form, we
develop theories based on them and on the way they are essentially
related to each other. Furthermore, in a metamathematical level, we are
able to posit new mathematical entities (negative roots, fractions, and
more) and formulate axioms based on them, or alter axioms in order to
explore mathematical consistency (as when the so-called “axiom of the
parallels” was not used and mathematicians could discover consistent
non-euclidean spaces). Such activity can be carried on as long as
logical consistency is preserved.
Hence, we have reached with Husserl a
satisfactory platonist epistemology of mathematics.
2.4
— Scientific Evidence for Mathematical Intuition
The evidence for mathematical intuition, meaning categorial intuition
and categorial abstraction, is evident everywhere in the world, in
every civilization. It is not a mystical intuition, but the result of
acts carried out by our mind in order to constitute objects and
states-of-affairs. George Ifrah shows how categorial intuition
manifests in animals:
Some
animal species possess some kind of notion of number. At a rudimentary
level, they can distinguish concrete quantities (an ability that must
be differentiated from the ability to count numbers in abstract). For
what of a better term we will call animals' basic number-recognition
the sense of number. [. . .]
Domesticated
animals (for instance, dogs, cats, monkeys, elephants) notice straight
away if one item is missing from a small set of familiar objects. In
some species, mothers show by their behaviour that they know if they
are missing one or more than one of their litter. A sense of number is
marginally present in such reactions. The animal possesses a natural
disposition to recognise that a small set seen for a second time has
undergone a numerical change.
Some birds have
shown that they can be trained to recognise more precise quantities.
Goldfinches, when trained to choose between two different piles of
seed, usually manage to distinguish successfully between three and one,
three and two, four and two, four and three, and six and three.
Even more striking is the untutored ability of nightingales, magpies,
and crows to distinguish between concrete sets ranging from one to
three or four.
[. . .]
What we see in domesticated animals is the rudimentary perception of
equivalence and non-equivalence between sets, but only in respect of
numerically small sets. In goldfinches, there is something more than
just a perception of equivalence — there seems to be a sense of “more
than” and “less then”. Once trained, these birds seem to have a
perception or intensity, halfway, between a perception of quantity
(which requires an ability to numerate beyond a certain point) and a
perception of quality. However, it only works for goldfinches when the
“moreness” or “lessness” is quite large; the bird will almost always
confuse five and four, seven and five, eight and six, ten and six. In
other words, goldfinches can recognise differences of intensity if they
are large enough, but not otherwise.
Crows
have rather greater abilities: they can recognise equivalence and
non-equivalence, they have considerable powers of memory, and they can
perceive the relative magnitudes of two sets of the same kind separated
in time and space. Obviously, crows do not count in the sense that we
do, since in the absence of any generalising or abstracting capacity
they cannot conceive any “absolute quantity”. But they do manage to
distinguish concrete quantities. They do therefore seem to have a basic
number sense.
114
From a husserlian standpoint, the term “number sense” is equivocal,
since we can see categorial objectualities that are not necessarily
numerical. There are different categorial forms being considered, like
“more than”, “less than”, and “sets”.. We should remember that no
categorial form is reducible to the other, and the notion of sets is
not reducible to the notion of number, and vice-versa.
115
This “number sense” only describes the ability animals have
of
conceiving objects as groups (sets). The data provided by Ifrah only
makes sense if we take into consideration that animals perceive
categorially because these categorial forms are founded on the sensible
objects in the states-of-affairs constituted by their mind.
However, we must recognize that depending on the animals' mental
faculties, its perception is limited by its ability to grasp certain
quantities of objects. This is not different with humans. Ifrah carries
out an experiment which I essentially reproduce in Illustration 3. Try
to know how many objects there are in each frame in Illustration 3
without counting them, just by glancing at them.
Illustration 3
The reader may have noticed that he or she can grasp the number of
objects just by glancing at one, two, three or four elements in a
frame. But if there are five or more objects it is almost impossible to
know right away the right amount without mentally grouping or counting
them. Since animals do not know how to count, they have a very limited
capacity for numbering. This not only applies to animals, but in some
human civilizations around the world, some people are not able to count
beyond three or four. For example,the Murray islanders use numbers
“one”, “two”, “three”, or , “four” but above that they call “a crowd of
. . .”. The same thing happens with Torres Straits islanders, among
other isolated civilizations.
116
Even babies have this mental ability of constituting states-of-affairs
through mixed categorial acts. For instance, Karen Wynn has
experimented with five-month-old babies and found that they can perform
elementary forms of mental arithmetic. Steven Pinker describes this
process:
In Wynn's
experiment,
the babies were shown a rubber Mickey Mouse doll on a stage until their
little eyes wandered. Then a screen came up, and a prancing hand
visibly reached out from behind a curtain and placed a second Mickey
Mouse behind the screen. When some screen was removed, if there were
two Mickey Mouses visible (something the babies had never actually
seen), the babies looked for only a few moments. But if there was only
one doll, the babies were captivated—even though this was exactly the
scene that had bored them before the screen was put into place. Wynn
also tested a second group of babies, and this time, after the screen
came up to obscure a pair of dolls, a hand visibly reached behind the
screen and removed one of them. If the screen fell to reveal a single
Mickey, the babies looked briefly; if it revealed the old scene with
two, the babies had more trouble tearing themselves away. The babies
must have been keeping track of how many dolls were behind the screen,
updating their counts as dolls were added or subtracted. If the number
inexplicably departed from what they expected, they scrutinized the
scene, as if searching for some explanation.
117
Even a similar methodology to this one
has shown that five-days-old babies are “sensitive to numbers”.
118
Husserl would say “Of course, they are sensitive to numbers!
Their consciousness is constituting states-of-affairs, because that is
what
is
originally given. They can see the sensible objects, and simultaneously
the categorial forms founded on them.”
In our case, we have reached a level of categorial abstraction, which
makes us separate sensible experience from formal judgments constituted
by formal categories. We do not need to perceive 99,999+1 sensible
objects (which would be impossible in light of our recent experiment)
to know that it
must
equal
100,000. Here we see our eidetic intuition in action when we recognize
the necessary relationship between three different numbers (99,999 + 1
= 100,000). In fact, we are able to see that we can even conceive all
kinds of mathematical entities, many of them not grasped on sensible
basis like irrational numbers, imaginary numbers (negative roots),
among others. These are constituted in light of mathematics’ third
stratum, where the mathematician is free to posit any mathematical
objects whatsoever, formulate axioms, and explore them as long as
consistency is preserved.
Hence, categorial
intuition, categorial abstraction, and eidetic intuition are perfectly
natural activities of the mind.
2.5
— Platonism and the Fallibility of Knowledge
One of the least understood aspects of platonism is that of the
fallibility of mathematical knowledge. For many antiplatonists, the
infallibility of truths and all the logical and objectual relationships
of abstract
entities
of all kinds implies necessarily infallibility of logical and
mathematical
knowledge.
Also they make the mistake of equating
a priori knowledge
with the
infallibility
of
a priori
knowledge. This constitutes a
non
sequitur,
it somehow implies that for an infallible mathematical or logical
proposition to exist, it follows that our knowledge about it has to be
infallible. It would be like saying that Uranus never existed for
millenia because humans never knew it. If we take it to the extreme
suggested by antiplatonists, Uranus still would not exist because our
knowledge of it is limited and not infallible.
In fact, for platonists, logic and
mathematics are science in the original sense of the word “
episteme”
in Greek. They provide formal knowledge. The way we grasp it is
completely different from that of natural science. For example, in the
case of natural science it is not enough to formulate consistent
theories,
we need to
find the empirical correlates
that might indicate that our theories are indeed correct. But logic and
mathematics do not proceed in such a way. Let us see two very simple,
but clear, examples, one of logic and one of mathematics.
Theorem. Modus Ponens preserves truth; so
if α and α→β are true in an interpretation I, then β has to be true in I.
Proof. Let us
suppose that α and α→β are true in I,
but that β is false in I.
If α is true and β false in I,
then that would mean that α→β should be false according to the
definition of implication. This, however, would contradict the
theorem's position that α→β is true. Therefore β should be true in I.
Notice
that in the case of logic we do not have to appeal to experience in any
way, its truth relies solely in its own definitions, rules, and axioms.
Let us see now an example in mathematics:
Theorem. There is no rational number c
which satisfies c2=2, so √2 is an irrational
number.
Proof. An irrational
number is that which cannot be expressed as a fraction (a ratio).
Suppose that there is a fraction
p/
q reduced to its
lowest terms, in such a way that
p2=2q2
This means that regardless of
q's
numerical value,
p
has to be an even number. Therefore, the value of
p is 2
a. If we substitute
p for
2
a then we
will obtain the following equation:
(2a)2=2q2
Therefore:
2q2=4a2
q2=2a2
This means that
q is an even number. If
p is even and
q is even, then
that means that they can both be divisible by two, so the fraction
p/q
is not reduced to the lowest terms. This would contradict the
hypothesis of the theorem that states that the fraction is reduced to
the lowest terms. Hence, there is no rational number
c which satisfies
c2=2.
We can see here that there is no appeal at all to sensible experience
in order to prove that √2 is an irrational number.
For platonists, both of these theorems are true and have always been
timelessly true. What we have done is not an “invention” or
“construction” of the truth of the theorems but a
discovery
of their absolute truth. The very notion of discovery implies these
truths have existed, but were not previously known, and they constitute
mathematical knowledge. If platonists argue the
discovery of such
objects, then the antiplatonists' view that platonism posits infallible
knowledge is invalid.
By the way, given that logic and mathematics as analytic disciplines
carry out their proofs in a completely different way than
natural-scientific theories as a body of synthetic judgments,
we can see a qualitative
difference between analytic and synthetic propositions.
If this is the case, Quine is wrong in denying the analytic/synthetic
dichotomy on the basis that there is no qualitative difference between
both kinds of statements. There is a difference in kind, and not merely
in degree of abstraction.
Now, we have to
confront a usual objection against platonists concerning mathematical
knowledge: the difficulty of proving certain theorems because their
proof is too long and sometimes some important factors are missing.
Philip Kitcher, for instance, presents this argument:
We
suppose, with the apriorist, that when we follow a proof we begin by
undergoing a process which is an a priori warrant belief in an axiom.
The process serves as a warrant for the belief so long as it is present
to mind. As we proceed with the proof, there comes a stage when we can
no longer keep the process and the subsequent reasoning present to
mind: we cannot attend to everything at once. In continuing beyond this
stage, we no longer believe the axiom on the basis of the original
warrant, but rather because we recall having apprehended its truth in
the approximate way. However, this new process of recollection although
it normally warrants belief in the axiom, does not provide an a priori
warrant for the belief. So, when we follow long proofs we lose our a
priori warrants for their beginnings.
119
But this argument does not refute the
fact that logic and mathematics considered in themselves are
a priori, nor does
it refute the statement that we have some
a priori
knowledge. All this argument shows is that our knowledge and
psychological processes are limited and that we are not fully able to
grasp the truth of certain logical or mathematical propositions. But
through a revision of the deductive process using mathematical truths
known by all mathematicians, the flaws of such a “long proof” can be
shown, then see what went wrong, and, sometimes, how to make it right.
Sometimes there is no proof because the mathematical conjecture is
wrong, but we do not know yet that it is wrong. James R. Brown shows an
example of Euler's conjecture, which is a generalization of Fermat's
Last Theorem. It states the following:
[. . .] if
n ≥ 3, then fewer
than n nth powers cannot sum to an nth power. As a special case, this
means that there are no solutions {
w,
x,
y,
z} to equation
w4+
x4+
y4=
z4.
The conjecture was well tested by examples, and for about two centuries
was widely believed as [Fermat's Last Theorem]. However,
counter-examples have been found recently, for example, 2,682,440
4
+ 15,365,639
4 + 18,796,760
4
= 20,615,673
4.
120
So,
Euler's conjecture was always false, even when everyone thought it was
true. The point platonists wish to make is not that our knowledge of
the logical-mathematical realm is infallible, but that sometimes many
of these
a priori
truths can
be discovered either by proving a certain conjecture or providing its
refutation. But our beliefs in them being true or false have nothing to
do with their objective truth value.
2.6
— Where and Why Do Mistakes Occur?
Most of the mistakes made in mathematics are due to three main reasons:
121
- We
could formulate false mathematical conjectures, but not know that they
are actually false. This is the case of Euler's generalization of
Fermat's Last Theorem which we have shown above, or David Hilbert's
(and Husserl's) belief in the completeness of mathematics. In both
cases, it was the apodeictic certainty of mathematics that led to the
discovery of these conjectures' falsity. These refutations do not mean
that mathematics is doomed to be uncertain as fallible inventions of
humans, nor does it mean that because this sole aspect has a similarity
in conjecturing and refuting like in science, it belongs to natural
science.122
It is an exaggerated claim to say that because some of these
conjectures were false there are “disasters” in mathematics.123 On the contrary, due
to the refutation of these conjectures we have more certain knowledge
of mathematics: our knowledge about Euler's conjecture is now more
certain than before we knew it was false.
- The
use of wrong and naïve concepts is also a source of mathematical
mistakes. For example, some of the ancient Greeks associated numbers
with geometrical objects, and adhered to these concepts the notion of
perfection (circles, squares, or equilateral triangles as perfect
shapes). Of course, the number 3 cannot be identified with the
equilateral triangle, nor is 4 identified to the square, etc. Nor are
these numbers expressing perfection of any kind. Also, the
misconception of numbers as distances prevented many philosophers in
history to adopt negative numbers and negative roots, and regarded them
as contradictory to certain axioms of mathematics. Simultaneously, this
conception of numbers prevented many ancient mathematicians from
developing a theory of complex numbers.
- Incorrect
application of accepted principles can be a source of mistakes. This
phenomenon can be shown with a simple example. Let us suppose that x = 1. Let us
follow the rules of algebra and multiply both sides of the equation
with the same variable x:
x2
– 1 = x – 1
(x – 1) (x + 1) = x – 1
If we divide both
sides by x
– 1 the result will be:
x + 1
= 1
x + 1
– 1 = 1 – 1
x = 0
Therefore,
x = 1 means
x
= 0. This is a mathematical impossibility because x cannot be 1 and 0
simultaneously. Although this demonstration seems to follow correctly
all algebraic laws, in reality it does not. If the premise is that
x = 1, then
x – 1 would be
zero, and division by zero is forbidden in algebra. The mistake was to
divide both sides of the equation by
x – 1.
124
As we have seen, we must be careful not
to confuse platonism with certainty of knowledge.
Our knowledge of
abstract objects is indeed affected by our limits and fallibility, but
that does not mean that there is no
a priori
knowledge of logical and mathematical objects. Mathematical
conjectures, to attain absolute certainty, must be proved or refuted in
some apodeictic way; if they are not, then they are just that:
conjectures.
Footnotes:
59Dummett,
2002, p. 20. [
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60p.
408. [
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61p.
409. [
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62LI.
Vol. II. Inv. II. §7. See p. 16 of this book. [
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63“EPR”
refers to the Einstein-Podolsky-Rosen Paradox (“EPR Paradox” for
short). Albert Einstein, Boris Podolsky, and Nathan Rosen tried to
refute the claim made by quantum physicists that somehow the observers
have influence over quanta. They made their case the following way:
suppose there is a positronium and that it releases two photons which
travel to two opposite sides of the galaxy. Quantum theory suggests
that the photons would literally be anywhere within a probabilistic
path. However, there are two detectors on both sides of the galaxy, and
despite the fuzzy path of the photons, those two detectors will detect
both photons at exactly the right locations. Given that
faster-than-light communication among both photons is impossible, the
photons would “magically” know where and when each other is being
detected to locate each other at exactly the right positions. For them,
this shows that the Copenhagen interpretation of quantum phenomena is
wrong, since it leaves unexplained how does this happen. In other
words, quantum physics as formulated is not complete. The EPR machine
operates in a slightly different manner. Based on suggestions by
physicist David Bohm, this machine does not measure position but spin,
making and EPR-like experiment possible (Popper, 2000, pp. 14-25).
[
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64A
positronium is an atom consisting of an electron and a positron.
[
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65Brown,
1999, pp. 16-17. [
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66Bunge
(1997) originally used Minerva and Mafalda, but it is the same idea
anyway (p. 65). [
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67Bunge,
1997, p. 65. [
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68Bunge,
1997, p. 72. [
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69Quine,
1953, pp. 20-46. [
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70Bunge,
1997, pp. 75-76. [
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71Bunge's
position is paradoxic. On one hand, he accepts the existence of
conventions, which do belong to the cultural realm. On the other hand,
he says that no cultural abstract products exist beyond the brain. Then
conventions cannot exist in any real sense, because they are abstract.
The cultural realm is rejected also by Bunge, especially as a form of
rejecting Popper's World 3. [
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to Text]
72FA.
pp. 33-38. [
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73FA.
pp. 33-38, 47-51. [
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74p.
33. [
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75p.
33. [
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76Brown,
1999, p. 13. [
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77Brown,
1999, pp. 13, 15. [
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78Brown,
1999, p. 15. [
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79Brown,
1999, pp. 15-18. [
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80Brown,
1999, p. 14. [
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81Katz,
1998, p. 1. [
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82Katz,
1998, p. 24. [
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83Katz,
1998, pp. xxxii, 38. [
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84Katz,
1998, pp. 34-35. [
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85Katz,
1998, p. 36. [
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86Katz,
1998, pp. 36-41. [
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87Katz,
1998, pp. 39-40. [
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88Katz,
1998, p. 45. [
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89For
Frege, objects are saturated entities, and numbers are seen as logical
objects in his philosophy of mathematics (FC. pp. 17-18). [
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90Katz,
1998, pp. 48-51. [
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91Katz,
1998, pp. 51-58. [
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92Katz,
1998, p. 73. [
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93Katz,
1998, pp. 73-74. [
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94Katz,
1998, pp. 26-28, 54, 55. [
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to Text]
95For
instance, “the set of all my children” is an example of “impure sets”
(Katz, 1998, p. 133). [
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Text]
96I.
§28. [
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97LI.
Vol. II. Inv. VI. §48; EJ. §§58-60. [
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to Text]
98LI.
Vol. II. Inv. VI. §40. [
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99LI.
Vol. II. Inv. VI. §45. [
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Text]
100LI.
Vol. II. Inv. VI. §46. [
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101LI.
Vol. II. Inv. VI. §46. [
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102EJ.
§61. [
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103LI.
Vol. II. Inv. VI. §60. [
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104LI.
Vol. II. Inv. VI. §60. [
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105LI.
Vol. II. Inv. VI. §60. [
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106LI.
Vol. II. Inv. VI. §60. [
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to Text]
107LI.
Vol. II. Inv. VI. §§52, 60; I. §§3, 9-10. [
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to Text]
108LI.
Vol. I. §67; LI. Vol. II. Inv. VI. §§42-44, 55, 61, 63-64. [
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109LI.
Vol. II. Inv. VI. §63. [
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to Text]
110FTL.
§§7, 9; EJ. §§3, 11. [
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Text]
111LI.
Vol. I. §67; LI. Vol. II. Inv. IV. §§10-14; LI. Vol. II. Inv. VI. §59;
FTL. §§12-13. [
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112LI.
Vol. I. §68; LI. Vol. II. Inv. IV. §12; FTL. §§14-22. [
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113LI.
Vol. I. §§69-70; FTL. §28-36. [
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to Text]
114Ifrah,
2000, pp. 3-4. [
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115Husserl,
1994/2004, pp. 12-19. [
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116Ifrah,
2000, p. 6. [
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117Pinker,
1994, p. 59. See Wynn, 1992. [
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118Pinker,
1994, p. 59. [
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119Kitcher,
1984, pp. 44-45. [
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120Brown,
1999, p. 166. [
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121Brown,
1999, pp. 18-23. [
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122Kline,
1985, pp. 391-427. [
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123Kline,
1985, pp. 3-7. [
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124Bunch,
1982, p. 13. [
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