Chapter 3:
Formal Science and Natural Science
3.1 - The Quine-Putnam Theses3.2 - Quantum Logic3.3 - Refutation of the First Quine-Putnam Thesis 3.3.1 - About the Problem of Quantum Logic 3.3.2 - General Theory of Relativity and Non-Euclidean Geometry 3.3.3 - Chaos Theory and Mathematics 3.3.4 - The Paradox of Revisability3.4 - Reply to the Second Quine-Putnam Thesis
Having established natural science as dealing with the physical world,
and formal science as dealing with ideal categorial forms and their
relationships between them, we proceed to explore how any theory of
natural science can imply a revision in the formal science.
3.1 — The Quine-Putnam Theses
One of the most controversial subjects in philosophy of science has to
do with the underdetermination of natural science. Most people refer to
the Duhem-Quine Thesis as one of its foundations. It should be
mentioned here that
there is no such thing as the Duhem-Quine Thesis.
As famous as this “thesis” may be, Pierre Duhem and W. V. O. Quine held
different points of view concerning natural science and how its
theories affect other branches of knowledge.
This should be corrected because this term has been widely used in the
fields of epistemology and philosophy of science. Donald Gillies made
an excellent exposition about the similarities and differences
regarding Duhem's thesis and Quine's thesis. Pierre Duhem said that in
the case of physics an experiment can never condemn a hypothesis but a
whole theoretical group. That what is really put to the test is not
merely a hypothesis, but a whole bunch of hypotheses, laws and theories
that the tested hypothesis supposes.
125 This aspect of physics
does not extend
to other branches of science such as medicine and physiology, and
definitely does not extend to formal science.
126 Evidence of this is
the fact that he refused to think that the general theory of relativity
was legitimate, because its use of non-euclidean geometry goes against
our intuition that space is euclidean.
127
Quine holds a very different point of view in “Two Dogmas of
Empiricism”. He denied the distinction between analytic and synthetic
propositions, and, as a result, he concluded that there is no actual
distinction between formal science and natural science except in
degrees of abstraction. Since there is no distinction among both
sciences, all of the formal and natural posits are nothing more than
convenient fictions. Quine says the following in “Two Dogmas”:
If this view is right, it is misleading to speak of the empirical
content of an individual statement—especially if it is a statement at
all remote from the experiential periphery of the field. Furthermore it
becomes folly to seek a boundary between synthetic statements, which
holds contingently on experience, and analytic statements, which hold
come what may. Any statement can be held true come what may, if we make
drastic enough adjustments elsewhere in the system. Even a statement
very close to the periphery can be held true in the face of
recalcitrant experience by pleading hallucination or by amending
certain statements of the kind called logical laws. Conversely, by the
same token, no statement is immune to revision. Revision even of the
logical law of the excluded middle has been proposed as a means of
simplifying quantum mechanics; and what difference is there in
principle between such a shift and the shift whereby Kepler superseded
Ptolemy, or Einstein Newton, or Darwin Aristotle? [. . .]
As an empiricist I continue to think of the conceptual scheme of
science as a tool, ultimately, for predicting future experience in
light of past experience. Physical objects are conceptually imported
into the situation as convenient intermediaries—not by definition in
terms of experience, but simply as irreducible posits comparable,
epistemologically, to the gods of Homer. For my part I do, qua lay
physicist, believe in physical objects and not Homer's gods; and I
consider it a scientific error to believe otherwise. But in point of
epistemological footing the physical objects and the gods differ only
in degree and not in kind. Both sorts of entities enter our conception
only as cultural posits. [. . .]
The over-all
algebra of rational and irrational numbers is underdetermined by the
algebra of rational numbers, but is smoother and more convenient; and
it includes the algebra of rational numbers as jagged on gerry-mandered
part. Total science, mathematical and natural and human, is similarly
but more extremely under-determined by experience. The edge of the
system must be kept squared with experience; the rest with all its
elaborate myths or fictions, has as its objective the simplicity of
laws.
128 As we can see, Quine's
proposal is much more radical than Duhem's, and it does extend to logic
and mathematics, specifically he presents the case of quantum logic.
Hilary Putnam is a bit more careful than Quine. He says that
propositions such as “2 + 2 = 4” without a doubt are true and not
subject to revision. However, he holds the belief that there are
mathematical propositions that are “quasi-empirical”, which may be
revised. Therefore there is no such thing as a priori knowledge.
129 He
seems to equate revisability with empirical experience, and in his mind
empiricism implies revision of supposed
a priori knowledge. We have argued in the previous chapter against Kitcher that
a prioricity
is not incompatible with revisability.
130 Also Putnam shows the example
of quantum logic as an instance of revisability in classic logic. He
says:
[in rejecting] the
traditional philosophical distinction between statements necessary in
some eternal sense and statements contingent in some eternal sense [. .
.] could some of the 'necessary truths' of logic ever turn out to be
false
for empirical reasons? I shall argue that the answer to this question is affirmative.
131 I
am inclined to think that the situation is not substantially different
in logic and mathematics. I believe that if I had the time I could
describe for you a case in which we could have a choice between
accepting a physical theory based upon non-standard logic, on the one
hand, and retaining standard logic and postulating hidden variables on
the other. In this case, too, the decision to retain the old logic is
not merely the decision to keep the meaning of certain words unchanged,
for it has physical and perhaps metaphysical consequences. In quantum
mechanics, for example, the customary interpretation says that an
electron does not have a definite position measurement; the position
measurement causes the electron to take on suddenly the property that
we call its 'position' (this is the so-called 'quantum jump'). Attempts
to work out a theory of quantum jumps and of measurement in quantum
mechanics have been notoriously unsuccessful to date. It has been
pointed out that it is entirely unnecessary to postulate the absence of
sharp values prior to measurement and the occurrence of quantum jumps,
if we are willing to regard quantum mechanics as a theory formalized
within a certain non-standard logic, the modular logic proposed in 1935
by Birkhoff and von Newmann, for precisely the purpose of formalizing
quantum mechanics.
132Therefore, we have here two
philosophers who seem to argue that natural science can indeed revise
formal science. Their statement can be summarized in two theses, which
shall be called here the Quine-Putnam Theses:
133- First Quine-Putnam Thesis: Mathematics and logic can be revised in light of recalcitrant experience as well as changes in scientific theories.
- Second Quine-Putnam Thesis: Mathematics exists because it is indispensable to science. This is the so-called indispensability argument.
Here I shall discuss both of these theses in light of the case they present and the refutation of such claims.
3.2 — Quantum Logic
One of the most cited cases we see in Quine's and Putnam's arguments
about the possibility of revising classic logic is the famous quantum
logic which has as its basis the empirical data on quantum behavior.
According to classic logic, this well-formed-formula, one distributive
law, is a tautology:
(α∧(β∨γ))↔((α∧β)∨(α∧γ))
If
this is a tautology, then it is always true, regardless of the truth
value assigned to the propositional variables. However, this logical
truth would not seem to hold in quantum physics. Martin Curd and
J. A. Cover give us an example of how this is so. Let us look at
Illustration 4:
Illustration 4
It
presents us the famous double-slit experiment. If we have a light
source on one side of a panel with two slits, we will be able to see an
interference pattern in a screen at the other end. If we conceive light
as a wave, we will be able to account for the interference pattern in
the screen. If we conceive light as made up of photons (light
particles), we are not unable to explain that phenomenon. At least
according to our experience with particles, the interference pattern
should not appear, we should see instead a set of two light bands
corresponding to the two slits in the panel. If we cover one of the
slits, the interference pattern is not shown anymore; if the two slits
are open, then it appears once again. How does the photon “know” that
the other is going to pass through the other slit to then form an
interference pattern? It might seem that the same photon can pass
through both slits to show the interference pattern.
Now, let us suppose that
p is the proposition “The photon is in region
R of the screen”,
q1 is the proposition “The photon went through slit
1”, and
q2 stands for the proposition “The photon went through slit 2”.
If the photon goes through slit 1 or slit 2, we do not see the
interference pattern formed in the screen. However, if they go through
both slits simultaneously, then we can see the pattern. Therefore, this means that:
p ∧ (q1 ∨ q2)
should not interderive with:
(p ∧ q1) ∨ (p ∧ q2)
This
is true in the quantum world, because a photon is assumed to pass
through both slits, not one.
134 Is this a refutation of classic logic
on empirical grounds, just as the Quine-Putnam Theses suggest?
3.3 — Refutation of the First Quine-Putnam Thesis
3.3.1 — About the Problem of Quantum Logic
One
of the ironies of philosophy is that one of the philosophers who
proposed the First Quine-Putnam Thesis was precisely the one who
refuted it: W. V. O. Quine. In his
Philosophy of Logic,
he discusses many cases of the so-called “deviant logics”, one of them
being quantum logic.
135 Many of those who favor and those who oppose
Quine pay no attention to this very important retraction from
statements he made in “Two Dogmas”.
For Quine, to deny a law of
logic or redefine logical connectives according to quantum phenomena
would be “changing the subject”. In classic logic, the meaning of
logical connectives is defined by their truth values. Quantum logic is
not truth functional and we cannot determine through truth tables
whether conjunction, disjunction, implication mean the same as in
classic logic. Therefore, quantum logic can hardly be considered a
refutation of classic logic. At most, it is an alternative logic, but
not a refutation of the older one.
136 He further says from his
pragmatic view:
[. . .] I would
cite again the maxim of minimum mutilation as a deterring
consideration. [. . .] let us not underestimate the price of a deviant
logic. There is a serious loss of simplicity, especially when the new
[quantum] logic is not even a many-valued truth-functional logic. And
there is a loss, still more serious, on the score of familiarity.
Consider again the case [. . .] of begging the question in an attempt
to defend classical negation. This only begins to illustrate the
handicap of having to think within a deviant logic. The price is
perhaps not quite prohibitive, but returns had better be good.
[.
. .] Now the present objection from quantum mechanics is in a way
reminiscent of this, though without the confusion. [. . .] Certainly a
scientist admits as significant many sentences that are not linked in a
distinctive way to any possible observations. He admits them so as to
round out the theory and make it simpler, just as the arithmetician
admits the irrational numbers so as to round out arithmetic and
simplify computation; just, also as the grammarian admits such
sentences as Carnap's 'This stone is thinking about Vienna' and
Russell's 'Quadruplicity drinks procrastination' so as to round out and
simplify the grammar. Other things being equal, the less such fat the
better; but when one begins to consider complicating logic to cut fat
from quantum physics, I can believe that other things are far from
equal. The fat must have been admirably serving its purpose of rounding
out a smooth theory, and it is rather to be excused than excited.
137
To add to this quinean rejection, many other philosophers and
scientists have objected to quantum logic for not helping us understand
what happens in the quantum world. For many of them, such a logic only
shifts the mystery from quantum physics to logic.
138 It is for this
reason that many philosophers of science as well as many scientists
themselves have completely rejected the Copenhagen interpretation.
Also, from an epistemological standpoint, this is one attempt to
translate quantum phenomena given in experience (
a posteriori), and turn it into some sort of
a priori
logical laws of quanta. The problem relies in the fact that, ideally
speaking, there can be numerous possible explanations for that
phenomenon, some which may not have been formulated yet. Those who
favor quantum logic only rely
on one particular interpretation
of quantum phenomena, and turn what apparently is a contradiction in
light of classic logic into a new kind of logic. The fallacy of this
procedure can be seen clearly once we realize that this logic is only
valid within a particular
a posteriori
natural-scientific theory. If in the future such a theory is refuted or
abandoned, this new logic will cease to be valid. However,
historically, classic logical laws remain true regardless of which
a posteriori natural-scientific theories are adopted.
Finally, we must not forget that the disjunction in the distributive
laws is an inclusive disjunction: or one, or the other, or both.
The norm in science is never to change or revise the axioms and
theorems of formal sciences on empirical grounds, but to change the
natural-scientific theory so it is consistent with logico-mathematical
laws and truths. There is a very simple but good example given by Carl
G. Hempel about this:
[. . .]
consider now a simple “hypothesis” from arithmetic 3+2=5. If this is
actually an empirical generalization of the past experiences, then it
must be possible to state what kind of evidence would oblige us to
concede the hypothesis was not generally true after all. If any
disconfirming evidence for the given proposition can be thought of, the
following illustration might well be typical of it: We place some
microbes on a slide, putting down first three of them and then another
two. Afterward we count all the microbes to test whether in this
instance 3 and 2 actually added up to 5. Suppose now what we counted 6
microbes altogether. Would we consider this an empirical
disconfirmation of the given proposition, or at least as a proof that
it does not apply to microbes? Clearly not; rather, we would assume we
had made a mistake in counting or that one of the microbes had split in
two between the first and second count. But under
no circumstances could the phenomenon just described invalidate the arithmetical proposition in question.
139This
illustrates very well the relation between formal sciences and natural
sciences. Some events in natural sciences seem to revise mathematics,
when in reality they do not. Let us see two more cases where this only
seems to happen.
3.3.2 — General Theory of Relativity and Non-Euclidean Geometry
One of the most cited arguments in favor of revision of mathematics is
the general theory of relativity and its adoption of non-euclidean
geometry. It can be said, for instance, that Einstein's discovery of
physical space-time being non-euclidean refuted euclidean geometry.
140
However, we must look carefully at these claims to understand well what
Einstein
really did in the case of non-euclidean geometry and his theories of relativity.
To be able to understand well what happens in the relationship between
geometry in general and the general theory of relativity, we should
examine the reason why non-euclidean geometry was developed. One of the
most significant axioms in euclidean geometry was the so-called “axiom
of the parallels” which states that given a line and a non-collinear
point, there is one and only one line that goes through that point
which is parallel to the given line. For many mathematicians, this
axiom was self-evident, but for others it was not. The same assumptions
were necessary for the proof of the theorem that stated that the sum of
the angles of a triangle is 180°. The more mathematics became an
abstract and rigorous discipline, the more the supposed self-evidence
of the axiom and the validity of the theorem were questioned. In the
eighteenth and nineteenth centuries, there was a conviction among some
mathematicians that to deny such an “axiom” would not lead to any
contradictions.
The Jesuit priest Gerolamo
Saccheri (1667-1733), after trying to prove the axiom of the parallels,
discovered accidentally that consistent non-euclidean geometry is
possible. He could not show through the method of
Reductio ad Absurdum
that the negation of the axiom of the parallels was false. So, without
knowing it, he showed that non-euclidean geometry could be consistent
and that it could be perfectly possible to conceive the angles of a
triangle being less than 180°. He rejected this conclusion on intuitive
grounds, but later Carl Friedrich Gauss (1777-1855) realized that
non-euclidean geometries are as valid as the euclidean one.
It was not until János Bolyai (1802-1860) and Nikolai Lobachevsky
(1793-1856) that a variant of non-euclidean geometry called “hyperbolic
geometry” was developed, which was ignored and rejected by most of the
other mathematicians at the time for being counterintuitive. This
hyperbolic geometry denied the axiom of the parallels and assumed, not
a flat kind of space, but pseudo-spherical. In it, the sum of the
angles of the triangle is less than 180°, it was possible to “draw”
more lines going through non-collinear points parallel to a given line.
Another mathematician, Bernard Riemann (1826-1866), developed another
kind of non-euclidean geometry called “elliptic geometry”, where the
sum of the angles of a triangle is greater than 180°, and where the
shortest distance between two points lies in a great circle (the line
which divides the sphere in two halves). It also makes possible for
more than one line to pass through two points.
As we can see, these non-euclidean geometries
were not grasped by experience in any way,
they came out as a direct result of centuries of reflections by
mathematicians, especially those in the eighteenth and nineteenth
centuries. Such theories did revise mathematics
but did not refute euclidean geometry.
In fact, euclidean space, under this conception, became one of an
infinity of possible mathematical spaces. It did not refute at all that
in euclidean space the sum of the angles of the triangle is 180°, or
that the Pythagorean Theorem is true. What it did refute was the
conception that the only valid geometry is euclidean geometry.
Natural science had nothing to do with this revision.
Then
what did Einstein do? He was acquainted with the philosophy of the
famous mathematician Henri Poincaré, who is considered today one of the
fathers of the general theory of relativity. Poincaré accepted the
mathematical validity of non-euclidean geometry.
141 For him, a
non-euclidean world is perfectly possible and is very different from
euclidean space,
142 but he further states the following:
It
is seen that experiment plays a considerable rôle in the genesis of
geometry; but it would be a mistake to conclude from that geometry, is,
even part, an experimental science [. . .] [Geometry] is not concerned
with natural solids: its object is certain ideal solids, absolutely
invariable, which are put a greatly simplified and very remote image of
them. The concept of these ideal bodies is entirely mental, and
experiment is but the opportunity which enables us to reach the idea.
The object of geometry is the study of a particular “group”; but the
general concept of group pre-exists in our minds, at least potentially.
It is imposed on us not as a form of our sensitiveness, but as a form
of our understanding;
only, from
among all possible groups, we must choose one that will be the
standard, so to speak, to which we shall refer natural phenomena.
Experiment guides us in this choice, which it does not impose on us. It tells us not what is the truest,
but what is the most convenient geometry.
It will be noticed that my description of these [non-euclidean] worlds
has required no language other than that of ordinary geometry. Then,
were we transported to those worlds, there would be no need to change
that language. Beings educated there would no doubt find it more
convenient to create a geometry different from ours, and better adapted
to their impressions; but as for us, in the presence of the same
impressions, it is certain that we should not find it more convenient
to make a change.
143 Poincaré's
statement regarding non-euclidean geometry is that it is as valid as
euclidean geometry, but it would not serve well at all to adopt
non-euclidean geometry as a convention in this world. It is perfectly
conceivable that it would make sense
to adopt non-euclidean geometry as a way to make the theories about the world simpler,
even if non-euclidean geometry itself is not as simple as euclidean
geometry. But due to the fact that our world seems to be in euclidean
space, he rejects the possibility of the eventual adoption of
non-euclidean geometry to make scientific theories simpler.
This is where Einstein comes in. One of the mathematical consequences
of the Lorentz Transformations and the adoption of the independence of
the constancy of light’s speed with respect to all inertial reference
frames is that nothing can travel faster than the speed of light. This
left a significant problem: according to newtonian mechanics, the
effect of gravity among massive objects is instantaneous, and there is
no account for Lorentz's spatial contraction. Poincaré influenced
Einstein by letting him see that it was possible to adopt a more
complicated mathematical model in order to simplify a scientific
theory. Due to the special theory of relativity we cannot talk about
rigid bodies, i.e. bodies in which the relative length, time frame, and
mass are not affected due to its speed with respect to other inertial
reference frames. Euclidean geometry would be an inappropriate model to
build a theory using special relativity. So, he had two options:
- To
retain euclidean geometry as the mathematical model for simplicity,
which would mean sacrificing the simplicity of the scientific theory
about space-time behavior.
- To adopt a more complicated
non-euclidean geometry as a mathematical model, but having the benefit
of a simpler scientific theory.144
By choosing the latter,
Einstein not only formulated a very consistent general theory of
relativity, but also was able to predict and include a series of
phenomena which were not accounted for in classic newtonian mechanics:
the Second Twin Paradox, the way light deviates near massive objects,
the motion of Mercury's Perihelion, and the Doppler Effect.
145
So, what we see here in the case of general theory of relativity is not
that it refuted euclidean geometry. Euclidean geometry itself does not
contradict non-euclidean geometry, because an euclidean space is one of
an infinity of possible spaces. Non-euclidean geometry came to be
because of internal problem solving processes within mathematics
itself, and its historical origin has nothing to do with its adoption
or rejection within natural science.
Therefore,
the general theory of relativity did not revise mathematics
at all. Quite the contrary. Einstein chose another mathematical model
of space which was available thanks to mathematicians' development of
non-euclidean geometry the previous century. He formulated the simplest
theory when he considered the mathematical implications of the special
theory of relativity as well as other phenomena. Instead of natural
science revising mathematics,
it was mathematics the field that revised natural science.
3.3.3 — Chaos Theory and Mathematics
Recently
there has been an enthusiasm about chaos theory, even to the point of
the absurd, mostly as a result of propaganda within the academy.
146
Some have presented chaos theory as a refutation of mathematical
method in general. What is chaos theory? Alan Sokal and Jean Bricmont
explain this very well:
What is
chaos theory about? There are many physical phenomena governed by
deterministic laws, and therefore, predictable in principle, which are
nevertheless unpredictable in practice because of their “sensitivity to
initial conditions”. This means that two systems obeying the same laws
may, at some moment in time, be in very similar (but not identical)
states and yet, after a brief lapse of time, find themselves in very
different states. This phenomenon is expressed figuratively by saying
that a butterfly flapping its wings today in Madagascar could provoke a
hurricane three weeks from now in Florida. Of course, the butterfly by
itself doesn't do much. But if one compares the two systems constituted
by Earth's atmosphere with and without the flap of the butterfly's
wings, the result three weeks from now may be very different (a
hurricane or not). One consequence of this is that we do not expect to
be able to predict the weather more than a few weeks ahead. Indeed, one
would have to take into account such a vast quantity of data, and with
such precision, that even the largest conceivable computers could not
begin to cope.
147 This is illustrated
best with what occurred to one of the fathers of chaos theory, Edward
Lorenz. He discovered what would later be known as the “butterfly
effect”, especially in relation to the weather. He used attractors in
order to describe the behavior of certain systems. Chaotic behavior, in
chaos theory, does not mean just pure disorder or pure randomness. A
system is chaotic if it depends greatly in the sensitivity of initial
conditions. Lorenz, studying the weather and picking up data,
eventually showed with an attractor, the behavior of a chaotic system.
Illustration 5 is a display of what has come to be known as the
Lorenz attractor.
Illustration 5
The attractor shows that even in what appears to be disorderly
behavior, there is an inherent structure within the stream of data,
even when the trajectories never intersect, and the system never
repeats itself. Such a view of chaotic patterns has been very useful in
explaining, for instance, Jupiter's Red Spot, which is essentially a
self-organizing system within a chaotic system. This seems to apply,
not only to nature, but also to economy, population growth, and so
on.
148 Chaos theory also includes the idea of
the fractal aspect of such behavior. Among many scientists and
mathematicians, it was Benoit Mandelbrot who discovered accidentally
that a diagram of income distribution can be correlated with the
diagram of eight years of cotton prices. Gleick tells more of this
story:
[. . .] when Mandelbrot
sifted the cotton-price data through IBM's computers, he found the
astonishing results he was seeking. The numbers that produced
aberrations from the point of view of normal distribution produced
symmetry from the point of view of scaling. Each particular price
change was random and unpredictable. But the sequence of changes was
independent of scale: curves for daily price changes and monthly price
changes matched perfectly. Incredibly, analyzed Mandelbrot's way, the
degree of variation had remained constant over a tumultuous sixty-year
period that saw two World Wars and a depression.
149
So, we see not only that there is an order within the pattern as
discovered by Lorenz, but also, in a sense, there was a correlation
between the whole and the parts of a chaotic system. Mandelbrot could
calculate the fractional dimensions of real objects according to shape
or any other irregular patterns. And it does not matter which
dimensional fraction we reduce it to, we are able to see that irregular
pattern again and again. He created the word “fractal” to refer to
these fractional dimensions and “fractal geometry”, to create the
discipline which has fractals as its objects of study.
This non-conventional way of looking at the world and the creation of
such a mathematical discipline was a very important step in
understanding the behavior of the physical world. Up to now we have
seen chaotic systems and fractals and their relationship with the
physical world. What about the mathematical realm? Many of these views
apparently also apply to pure mathematical objects, such as the very
well known Mandelbrot Set. Gullberg explains in full detail what the
Mandelbrot Set is:
The
fractal behavior in the complex number plane is demonstrated by
iterating a nonlinear function whose variables include its own result.
If a set of an infinite sequence f(z), f(f(z)), f(f(f(z)))..., where z is a complex number, is plotted on a graph, the sequence of iterates may
- be unbounded; or
- jump around within a bounded region
If (2) holds, we say that
z lies in the “filled-in Julia set for
f” [. . .]
The Mandelbrot set is related to the Julia set, but for it the defining variable is the
c in
f(
z)=
z²+
c, where
z and
c are complex numbers. Starting with
z=0+0
i, we look for the complex numbers
c, such that 0,
f(0),
f[
f(0)], ... remain bounded.
If we let
z=0+0
i in
f(
z)=
z²+
c, then
f(z) = f(0+0i) = (0+0i)2 + c = c
f(f(z)) = f(c) = c2 + c
f(f(f(z))) = f(c2 + c) = (c2 + c)2 + c
and the process may be iterated
ad infinitum. [. . .]
We are now in a position to define the
Mandelbrot set as the set of complex numbers c for which the iterated
f(
z)=
z²+
c remains bounded. The initial value of
z is 0+0
i and each subsequent value of
z is used to find the next one.
150A more formal definition of the Mandelbrot set is the following:
Definition. The Mandelbrot Set
M is the set of complex numbers
c such that the sequence
does not approach ∞ as
n gets larger.
Through the process of iteration in a computer program, we can actually
produce Illustration 6, which is the graphical representation of the
Mandelbrot Set.
Illustration 6
The
image, as complex and irregular as it may appear, has also in it an
intrinsic pattern. Using a certain sequence of finding fractions on the
Mandelbrot Set, we are able to find again a fraction that contains once
more its original image. In Illustration 7, the white rectangles show
how we can “zoom in” the Mandelbrot Set, and find gradually the same
image pattern of the Mandelbrot Set. If we repeat that process, we will
find another image of the Mandelbrot Set, and so on.
Illustration 7
So, it seems that once again we are apparently faced with a possibility
that empirical sciences have revised mathematics. The reasoning we
apply in empirical experience is the same we apply to mathematical
objects, because fractal reasoning applies to objects of sensible
experience and also to certain mathematical objects. This reasoning
itself is based on experience. Therefore, on the basis of sensible
experience, mathematical reasoning has been revised.
To be able
to understand this well, we have to distinguish two very important
sides of chaos theory, one that has to do with natural and empirical
sciences, and another solely having to do with pure mathematics. Some
people have alleged that chaos theory adds uncertainty to mathematics,
especially on the basis of experience. First, I wish to ask: if
mathematics is uncertain on the basis of experience, then what
guarantees the certainty of chaos theory in the first place? If chaos
theory in a sense refutes mathematical certainty, then why does it use
mathematically certain rules to be able to understand this with
certainty?
Alan Sokal and Jean Bricmont have pointed out the
confusion of those who argue this way. They confuse determinism with
predictability, but this aspect applies only to empirical sciences, not
to pure mathematics. Chaos theory does not refute determinism at all,
all chaotic systems in the world are determined according to physical
laws. They just have the peculiar aspect of its determination depending
on the sensitivity of initial conditions. However, due to the fact that
we are not able to account for each and every single variable
that intervenes in a chaotic system, we are not able to predict with
100% accuracy certain phenomena. That is why Edward Lorenz discovered
that we are able to predict the weather only in short term within a
statistical model, and such a prediction loses its certainty as time
goes by.
151 But none of this revises mathematics at all, on the
contrary, as I will show, the application of certain mathematical
chaotic notions such as strange attractors or fractals, is no different
from the application of non-euclidean geometry to physics. So, for this
discussion we will have to look at the part of chaos theory that deals
with mathematics.
In general, there are some misuses of the
notion of fractals that should be mentioned here. Notice that many of
the advocates for fractal geometry say that fractals are a more
accurate representation of reality. As Gleick said about Mandelbrot:
Clouds
are not spheres, Mandelbrot is fond of saying. Mountains are not cones.
Lightning does not travel in a straight line. The new [fractal]
geometry mirrors a universe that is rough, not rounded, scabrous, not
smooth. It is a geometry of the pitted, pocked, and broken up, the
twisted, tangled, intertwined.
152The classic way in which
Mandelbrot confronts a certain problem concerning the usual way we do
geometry, is to ask: “How long is the Coast of Britain?” Mandelbrot
found out that Lewis F. Richardson saw discrepancies about 20% in the
estimated lengths of the coasts of Spain, Portugal, Belgium, and the
Netherlands. Mandelbrot wanted to take another approach: the fractal
approach. He arrived to the conclusion that a coastline is infinitely
long. In theory, if we continue “zooming in” the coastline, not only
will everyone discover how long it is, but also at one point, we will
see the same irregular shapes as the original (as the famous fractal
shore illustrates).
153 But is this true?
Gullberg comments about this:
The
universe is replete with shapes that repeat themselves on different
scales within the same object. In Mandelbrot's terminology, such
objects are said to be
self-similar.
In the idealized world of
mathematics, there are several well defined figures that are
self-similar and an infinite number of such figures may be generated
through iteration of functions [. . .] The word fractal — coined by
Mandelbrot — was intended to describe a dimension that could not be
expressed as an integer, today, “fractal” is generally understood to
mean a set that is self-similar under magnification.
Unlike
mathematical fractals, no object in nature can be magnified an infinite
number of times and still present the same shape of every detail in
successive magnifications – one reason being the finite size of
molecules and atoms. Yet fractal models may provide useful
approximations of reality over a finite range of scales.
Mandelbrot
and others have applied fractals
as explanatory models of natural
phenomena involving irregularities on different size scales. This
technique is used in graphical analysis in such diverse fields as fluid
mechanics, economics, and linguistics and the study of crystal
formation, vascular networks in biological tissue, and population
growth.
154The word “models” here is key to understand exactly
what is going on. As we can see, fractal geometry is as much an
approximation to reality as euclidean and non-euclidean geometries are.
What chaos theorists do essentially is use fractal geometry and many
other mathematical notions and apply them to experience, the very same
way non-euclidean geometry was applied to simplify scientific theory
and explain the physical world more accurately. In this way, fractal
geometry helps us to understand better the chaotic systems in the
world. None of this refutes the validity of euclidean geometry at all
nor any other mathematical axiom or theorem. The choice of the
mathematical models depend greatly on which kind of natural-scientific
theory we choose, how that model is pertinent to it, and if it really
simplifies the theory in such a way that actually makes a better
understanding of the world possible.
So, attempts to make chaos
theory a way to revise formal science basing ourselves on empirical
experience is also doomed to failure.
3.3.4 — The Paradox of Revisability
As
we said in
Chapter 2, we cannot forget the paradox of revisability
pointed out by Jerrold Katz. This is a big headache to all of those who
believe in the First Quine-Putnam Thesis, because this paradox
discovers two facts: the absolute necessity of some basic logical
truths such as the principle of no-contradiction, and it also discovers
the inviability of the potential revision of
all formal science.
3.4 — Reply to Second Quine-Putnam Thesis
The
Second Quine-Putnam Thesis has been known as the “indispensability
argument”, which states that mathematics is meaningful because of the
fact that it is indispensable to science. This is related to the First
Quine-Putnam Thesis, that somehow formal sciences are revised in light
of recalcitrant experience. I wish to point out the indispensability
argument as a strange claim.
Rosado Haddock says that if
mathematics is subordinated to physics, it is strange that mathematics
does not refer at all to physical entities or theories of any kind. In
fact, it seems that contrary, to physical theories, many mathematical
truths are self-evident and true in every possible world.
Now,
although applicable to the physical (and other) sciences, mathematical
theorems seem to be true even if all actually accepted physical
theories were false and, thus, the claim that only after the advent of
modern physical science can we argue that mathematical theorems are
true seems really amazing, to say the least. It is also extremely
unreasonable to think that before the advent of modern physical science
there was no way to establish the existence of mathematical entities,
thus, e.g., that there exists an immediate successor of 3 in the
natural number series. Moreover, it is perfectly conceivable that there
exists a world in which all mathematical theorems known to present-day
mathematicians are true (supposing that current mathematics is
consistent), and that mathematicians know as much mathematics as they
actually know, but in which none of the physical laws accepted as true
nowadays were known to humanity. What is not possible is a world in
which physical science were as developed as it actually is, but in
which our present mathematical theories (especially those applicable to
present-day physical science) were not valid, or, at least, were not
considered to be valid.
155 Katz also made his criticism along
this line, stating that we can establish the existence of these
mathematical entities even without empirical science.
156 So, can we
remain with a straight face when we state that the validity of
mathematics depends on the validity of scientific theories? It seems
the other way around. We admit that these replies to the Second
Quine-Putnam Thesis are not a refutation per-se of the claims, but they
show how unlikely this thesis seems to be.
Footnotes:125Duhem, 1991, pp. 183-188; Gillies, 1993, pp. 98-99. [
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126Duhem, 1991, pp. 180-183. [
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127Curd & Cover, 1998, p. 377; Gillies, 1993, p. 105. [
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128Quine, 1953, pp. 43-45. [
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129Putnam, 1975, pp. 124-126. [
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130See Hale's (1987) comments on Putnam's view on revisability of
a priori disciplines (p. 143). [
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131Putnam, 1975, p. 174. [
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132Putnam, 1975, p. 248. [
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133This
is a phrase used by Katz (1998) to refer to Quine's and Putnam's
statement on the revisability of formal sciences in light of experience
(p. 50). [
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134Curd & Cover, 1998, p. 380. [
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135The
other “deviant logics” Quine (1970) mentions are intuitionistic logic
(which rejects the principle of excluded middle) and various
multi-valued logics. [
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136Quine, 1970, pp. 83-86. [
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137Quine, 1970, p. 86. [
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138Curd & Cover, 1998, p. 380. [
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139Hempel, 2001, p. 4, my italics. [
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140Putnam, 1975, pp. xv-xvi. [
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141Poincaré, 1952, p. 50. [
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142Poincaré, 1952, pp. 64-68. [
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143Poincaré, 1952, pp. 70-71, my italics. [
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144Einstein, 1983, pp. 33-35, 39. [
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145See also Carnap's comments in Reichenbach, 1958, p. v. [
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146I
need not emphasize the huge problems of certain “thinkers” who use
chaos theory to support the latest nonsense that comes to their mind.
Alan Sokal and Jean Bricmont illustrate very well the overwhelming
confusions concerning so-called “thinkers” like Jean-François Lyotard,
Jean Baudrillard, Gilles Deleuze, and Félix Guattari (Sokal &
Bricmont, 1999, pp. 147-168). For a very sober research on chaos
theory, Sokal and Bricmont have suggested the following readings:
Kadanoff, 1986; Matheson & Kirchoff, 1997; Ruelle, 1991; and Van
Peer, 1998. [
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147Sokal & Bricmont, 1998, p. 138. [
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148Gleick, 1987, p. 55. [
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149Gleick, 1987, p. 86. [
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150Gullberg, 1997, p. 633. [
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151Sokal & Bricmont, 1998, pp. 140-146. [
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152Gleick, 1987, p. 94. [
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153Gleick, 1987, pp. 94-96. [
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154Gullberg, 1997, p. 626, my italics. [
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155Hill & Rosado, 2000, p. 269. [
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156Katz, 1998, pp. 50-51. [
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