Friday, 22 June 2018
Friday, 17 February 2017
EFFECTIVENESS OF MATHEMATICS
RAJARAM
VENKATARAMANI
ABSTRACT
The beauty of Mathematics has only
been excelled by its effectiveness in Physics. Consequently, different schools
of Philosophy of Mathematics have reflected on the Ontology of Mathematical Objects and issues around Applicability and Explanatory Power, Indispensability
and Unreasonable Effectiveness. I analyse
these philosophical issues using Shapiro, Colyvan and Wigner’s Paper as the
main sources. I argue against the realist positions taken by Plato, Shapiro,
Putnam, and Maddy. I broadly agree with Wigner but take a refined Formalist
position to resolve the Wigner Puzzle, as Steiner terms it.
1.
ONTOLOGY – WHAT ARE
MATHEMATICAL OBJECTS?
A Mathematical Object is a well –
defined concept with which one can do deductive reasoning to come up with a
Mathematical Theorem. The examples of Mathematical objects are numbers,
functions, sets, groups, vector spaces and so on. As these apply to
Physics, I
review Metaphysical and Epistemological questions from the point of view Platonism,
Logicism, Nominalism, Structuralism and Formalism. I use Shapiro’s “Thinking
About Mathematics” as main reference to summarise these schools of thought for
the purpose of analysis.
Plato is a realist in truth value and
ontology. In Platonism, Mathematical Objects are abstract,
ideal and constant entities that belong to the world of being rather than the changing
world of becoming. I disagree with original Platonism because
a.
how we access the trans – space-time realm is resolved using non-falsifiable
propositions such as “recollection by an omniscient soul”
b.
it replaces known concrete
entities with an unknown ideal and
c.
it is speculative about whether Mathematics belongs to Forms or
Intermediaries. Why not Alice’s wonderland?
Gödel considers
the nature of abstract objects as “classes and concepts” which resolve “b”
(unknown ideal) but not “c” (speculative). Full blooded Platonism, which says
that every object that could exist does exist, overcomes “a” (access) but not “b”
(unknown ideal) or “c” (speculative). Given these difficulties, why would one
be a platonist (small p to indicate modern versions)? The primary motivations
are
I.
Physical Entities are finite
whereas Mathematical Objects are infinite and therefore must be transcendental
to space-time and
II.
Mathematical truths are
objective
On I, are
infinite Mathematical Objects conceived by the mind? If not, infinitude of Mathematical
Objects is only a mental concept and we don’t look for a shelf to store hare’s
horn. If yes, then there is no extra realm other than the mind required to
store these objects. Hilary Putnam subscribes
to a form of realism based on argument II. Now, is objectivity of Mathematics
known objectively or subjectively? I
argue that it is only known subjectively to Mathematicians who accept a common
axiomatic system. Those who were on either side of the fence on Euclidian
vs. Hyperbolic Geometry did not accept the same set of Mathematical truths. Therefore,
Mathematical truths are not objective.
John Bigelow
(1988) and Penelope Maddy (1990) argue for a Physicalist account of
Mathematical Objects. “Every time you
look in the refrigerator and see a dozen eggs you are seeing the set of 12
eggs. You are thus face to face with a Mathematical Object called the set.”1 Now, if we
break the eggs, do we break the concept of a set? No. Also, we don’t have the option
of not seeing dozen eggs when present but it takes conscious effort to see a
set where there are dozen eggs. Therefore, I argue that a set does not have
physical reality but is a product of cognitive function. Now, can we think of a
dozen eggs as null set? No. Therefore, I argue that Mathematical Objects though
mental concepts correlate to physical realities that have similar patterns in terms
of structure and / or behaviour. In summary, I disagree with all forms of Platonism
and am an anti – realist in both truth value and ontology.
According to
Structuralism, the subject of mathematics is the Relationships between Mathematical
Objects, not the Mathematical Objects themselves. Mathematical objects are just
places or roles in structures (Julius Caesar could be number 2 as long as he
relates to other natural numbers in the structure). In ante – rem structuralism
subscribed to by Shapiro and Resnik, structures are Platonic universals and therefore
have the problems “a” (access), “b” (unknown ideal) and “c” (speculative) discussed
above. Also, how can meaningful structures exist without their constituent
parts (Mathematical Objects)? In In – Re
structuralism, structures come in to existence when they are instantiated and
are Aristotelian universals. The problem, as Colyvan points out, is that it is
too modest and may not provide rich enough structures for Mathematics. For
example, real numbers need continuous and unbounded space when instantiated and
natural number need infinite objects. 2 I think the structures are what the human
mind creates based on the relationships between entities in the physical world
or mental realm. To give them any physical reality would lead to the kind
of problems we discussed in the case of Maddy’s egg-set.
According to
Logicism, Mathematics is reducible to Logic – deductive reasoning based on a
set of axioms. Frege considered some of Mathematics but not all as reducible to
Logic. He proposed that “abstract” Mathematical objects are the content of our
“thoughts”. I disagree with it because it only solves problem “b” (unknown
idea) but leaves “a” (access) and “c” (speculative) intact. Russell considered
all of Mathematics as reducible to Logic and took an anti-realist position. I arrive at the same conclusion as stated
above but disagree with him on his approach. He offered paradoxes as evidence
for human creation of Mathematics. Will resolution of paradoxes then imply that
Mathematics is not human creation? No, as new paradoxes may be posed and one
can never know if all possible paradoxes have been posed and resolved for an
axiomatic system.
In Formalism, Mathematics
is a mind or a computer game where symbols are manipulated using a set of rules
or an algorithm. The symbols are syntax without any inherent meaning. They
develop meaning only when bound to semantics. I think that this is the most reasonable
position because Mathematics is often done for fun as a game and for its
aesthetic value. Hilbert’s Formalism was in part thwarted by Gödel’s
Incompleteness Theorems and Modern Formalism tries to revitalise it. Now, it
may not be possible to resolve the tension between completeness and consistency
but leaving that aside I agree with Thomae’s
remarks “the formal standpoint rids us of
all metaphysical difficulties”3 Wigner says, “…mathematics
is the science of skilful operations with concepts
and rules invented just for this purpose.”4 We see that his
definition is similar to a Formalist definition if we replace “concepts” with “symbols”,
which are only notations for concepts: “Mathematics
is the manipulation of the meaningless symbols of a first-order language
according to explicit, syntactical rules.”5
I think that the
Formalist view point is extremely useful in addressing the appropriateness of
Mathematics to Physics. The operator for Mathematics is Logic, the operands are
symbols and the outcomes are Mathematical Theorems that represent the causal
relationship between symbols (semantically overloaded with concepts). A Physicist observes Physical Entities and
discovers patterns that he or she formulates as a Physical Law which defines
the causal relationship between the Physical Entities. Once observed, a Physical
Entity is nothing more than a mental concept that can be represented by a symbol.
At that level, it is ontologically no different to a Mathematical Object. If a Mathematical Object is congruent to a
Physical Entity, that is equivalent in terms of its essential attributes, how
can it be different for the purpose of analysis? If the analysis-rule-set (Algorithm
or Mathematical Mechanism) is congruent to the Physical Process, then it will
do to Mathematical Objects what the Physical process will do to Physical
Entities. Therefore, a Mathematical
Model (Mathematical Objects plus Mechanism) is non-different to Physical System
(Physical Process plus Entities). Though essentially the same at mental
realm, for the sake of convention, we will continue treat Mathematics and
Physics as if they are two domains in.
2.
APPLICABILITY AND EXPLANATORY
POWER – HOW MATHEMATICS APPLIES TO AND EXPLAINS PHYSICS AND ITSELF?
Shapiro states
the problem of applicability of Mathematics to Physics (and other empirical
sciences) and the need for resolution as, “Given
the extensive interactions, the philosopher must at least begin with the
hypothesis that there is a relationship between the subject matter of
Mathematics (whatever that is) and the subject matter of Science (whatever that
is), and that it is no accident that Mathematics applies to material reality,
Any Philosophy of Mathematics or Philosophy of Science that does not provide an
account of this affiliation is incomplete at best”. Implicit in this
statement by Shapiro is the notion that Mathematics is distinct from Physical
reality but this is true only for ontological realists. For them, the problem of applicability is
mainly due to the access problem. In other words, how are we able to access the
abstract realm? For an anti-realist, the problematic abstract realm of real
concepts such as numbers, sets, functions and so forth do not exist. Therefore,
an anti-realist has one less problem to solve than a realist. However, Colyvan
points out that the problem of applicability is there for both realist and anti
- realist.6
What is the problem? Physics is driven by empirical evidence and Mathematics is in some
sense disconnected from the Physics but turns out to be more than just right
for Physics! Mathematics seems to proceed via apriori means and yet finds
applications in a posteriori science. As a result,
not only Wigner but many others have wondered about the relationship between
the two subjects – Paul Dyson, Steven Weinberg and Mark Steiner to name a
few. The realists have to solve ontology / access
and methodology problems while anti – realists have to solve the latter. Though
they have one less problem to solve, they cannot do it as long as they have
mind (Mathematics) and matter (Physics) duality as truths in one realm (mind)
happen to be in the other (matter). The realists tend to find a third realm of
which these two are concrete instances but they are inventing a fairy land
which is neither verifiable nor logically consistent. A simpler, more aesthetic
and reasonable solution is to admit of only one realm – the mental reality of
concepts represented by symbols. I am not suggesting that there is no physical
reality but that it is indeterminate and what is known about it is only a
concept in the mind represented by a symbol. Mathematics applies to Physics because the underlying indeterminate
reality produces logically related concepts (symbols) in the mind. A
concept has to be a cause or an effect or how can we even know it? Therefore,
Logic inheres in the universe. (It may of interest to note that if Miracles (M)
are effects that have no Logical cause (L) and God (G), by definition is
impartial (I), then G --> I, I --> L, L --> not M. Therefore, M -->
not L --> not I --> not G. Therefore, those who admit miracles should
admit that there is no God to perform them!)
Mathematics, not
only applies to Physics but also explains both Physics and itself (Mathematics)!
Colyvan defines explanation as a story that makes
something puzzling less puzzling. I would rather call it a statement that
produces an intuitive understanding in the human mind. Mathematical Explanation
is what makes both Mathematics and Physics comprehensible. I would like to
summarize Colyvan’s treatment7 and highlight one critical point
where I differ (causal explanation in Mathematics required for its
appropriateness). When Mathematics explains itself, it is called an intra –
Mathematical explanation. It is done through i. Reductions (e.g. number system
to set theory) ii. Generalisations (e.g. Groups as abstraction) and iii. Proofs
(By Contradiction, Exhaustion and Induction). Euler’s Formula, which is the
result of domain extension, provides deep insights in complex analysis,
trigonometry and exponentiation. Colyvan categorises proofs as Explanatory and
Non-Explanatory but is reluctant to attribute explanatoriness or lack thereof
to styles of proof. He suggests that the distinction between an explanatory vs.
non – explanatory proof is the difference between classically – valid proof vs.
relevantly – valid proof. When Mathematics explains Physics, he calls it an
extra – Mathematical explanation. For example, the Lorentz contraction is a
Mathematical (Geometric) explanation involving Minkowski space and metric, the
Kirkwood gaps in Asteroid Belts are explained by Eigen Values (Functional
Analysis), hexagonal structure of hive – bee honey comb is explained by the Honey
Comb Theorem and number theory explains why North American Cicadas have prime
number life cycle. What is the obstacle to unified theory of explanation? While
Mathematical Theorems such as Pythagorean and Fundamental Theorem of Algebra
are not causal explanations, Physical Laws are. It is on this point that I
disagree with Colyvan because there are Analytic, Topological, Algebraic and
Geometric proofs for these Theorems that explain why they hold. These are all causal explanations but don’t
appear to be so only because the sophistication of reasoning using manipulation
of symbols which eclipses logical (cause – effect) relationship between the
concepts (symbols).
Wigner cites
three examples of applicability8. The first is the parabola being
useful to explain the planetary motion and stone pelting. The second is matrix
mechanics explaining position and momentum variables of the equations of
classical mechanics for H2 and Helium atoms.
The third is the Quantum Electro Dynamics or theory of Lamb Shift, which
is a pure Mathematical Theory and the contribution of experiment was to show
the existence of measurable effect. I think explanation is that the Physicist
matches the logical patterns that he observes in the Physics to Mathematics
that he or she knows. There recurrence of patterns gives multiple opportunities
for Mathematics to apply to Physics. The
contribution from Physics to Mathematics substantiates this point that that
there is fundamental commonality and coherence between the two domains. As
Arthur Jaffe and Edward Witten say, “Mathematical
structures of importance have first appeared in physics before their
mathematical importance was fully recognized by Physicists.”9
The discovery of calculus was motivated by Newtonian mechanics. Quantum Field
Theory (QFT) has led to new Mathematical structures in Probability, Analysis,
Algebra, and Geometry. There are new non-Gaussian measures and
Euclidean-invariant measures on spaces of generalized functionals. The
Renormalization Theory which originally appeared in Quantum Electro Dynamics
(QED) as a method to treat infinities became rigorous Mathematics with wider
applications such as convergence of Fourier Series and Classical Dynamical
Systems.
3.
UNREASONABLE EFFECTIVENESS – is
mathematics more than just a precise language to describe physics? is it
unique?
Colyvan cites
Maxwell’s Equations of Electromagnetism as an example of Unreasonable Effectiveness
of Mathematics in not just describing Physics but catalysing the discovery of
new Laws of Physics. The equation states
that the curl of a magnetic field is proportional to the sum of conduction and
displacement currents. Maxwell found that Gauss’s Law of Electricity, Gauss’s
Law for Magnetism, Faraday’s Law and Ampere’s Law jointly contravened the
conservation of electric charge. He modified Ampere’s Law by introducing rate
of change of an electric field (displacement current) by drawing an analogy
from conservation of mass in Newton’s gravitation theory. Though originally
formulated for charges with constant motion, he assumed that they would hold
for charges with accelerated movement and systems with zero conduction current.
In this more general setting, it was possible to predict that a changing
magnetic field would produce a changing electric field and vice versa. He also
predicted that the interaction would produce electromagnetic radiation that propagates
at the speed of light. Are Mathematical equations wiser than even their
discoverers that we get more out of them than we put them as Freeman Dyson says? According to Mark Steiner,
the puzzle is not simply the extraordinary appropriateness of Mathematics for
formulation of Physical Laws but the role of Mathematics in the very discovery
of those Laws! This indeed is unreasonable effectiveness. Aesthetics, the
attraction to symmetry, fertility and simplicity, plays an important role in
the development of Mathematical Theorems and seems to be an integral part of
the process in discovery of Physical Laws as well!10
Wigner gives a number of examples to show
that Mathematics is Unreasonably Effective. Complex
Analysis which is fundamental to Modern Physics is based on the human rules for
manipulation of imaginary numbers! Riemann Geometry’s lays the foundation for
Einstein’s General Relativity to describe gravity. Functional Analysis and
Group Representation Theory are fundamental to Quantum Mechanics. In Quantum
Mechanics, states are vectors in Hilbert Space and observables are self –
adjoint operators on these vectors. The possible values of the observations are
the characteristic values of these operators11.
There are three arguments offered against Unreasonable
Effectiveness based on Hamming (1960) and I disagree with them all. First argument is, it is difficult to get Mathematical Models to
work in tandem with Physical world. It is by overlooking such difficulties and
looking at the end product that unreasonable effectiveness problem is posed. My
response is that the modelling difficulties are skill issues and not limitation
of Mathematics itself. It is not even always required to model and solve the
equations of a Physical system. Poincare Phase Diagrams represent the solutions
of Non – Linear Differential Equations that cannot be solved. I think it is a
good example of models providing deep insight about Physical Systems. Second
argument is the existence of corpus of unapplied Mathematics. Now, Boolean
Algebra was just a game but modern civilisation as we know can’t exist without
it. How can we know that unapplied Mathematics of today will not have stunning
applications in future? Also, even one example such as Riemann Geometry or
Hilbert Space is strong enough to say that Mathematics is unreasonably
effective. Third argument is, Mathematics is the only tool available to study
Physics and with the lens of Mathematics, a Physicist is likely to see the
parts of the world that are amenable to Mathematisation. My response is that we
are not able to come up effective theories (with predictive power) using poetry
or art though there is no shortage of poets and artists looking at the world. Therefore,
there is no denying the fact that Mathematics is Unreasonably Effective in
Physics. I call it Extremely
Effective rather than Unreasonably
Effective to show that it is not mystical in any sense but within the logical
cause – effect framework, which I have already argued earlier is the nature of reality
as we know it. Also, is the solution to this puzzle of Effectiveness in
Mathematics or Physics or both? In all the cases, Mathematics should be able to
explain it causally as we saw in the previous section removing any mystery
about it. Therefore, it is extremely effective but there is no mystery as
Wigner may argue.
Now, is
Mathematics uniquely appropriate? There are two main points in Eugene Wigner’s
paper “… mathematical concepts turn up in
entirely unexpected circumstances. Moreover, they often permit an unexpectedly
close and accurate description of phenomena in these connections” and “we cannot know whether a theory (physical
law) formulated in terms of mathematical concepts is uniquely appropriate.”12
He decides not to focus on the second point in the paper and says
that we don’t have evidence of unique appropriateness. I would like to argue
that it is uniquely appropriate. Paraphrasing his problem, how do we know that,
if we made a theory [T1] which focusses its attention on phenomena [P1]
we disregard and disregards some of the phenomena now
commanding our attention [P2], that we could not build another theory
[T2] which has little in common with the present one
but which, nevertheless, explains just as many phenomena
[P1 OR (P1 AND P2)] as the present theory?” My
response is that if T1 and T2 are Mathematical Theorems based the same
axiomatic system A1, then they cannot be mutually contradictory as it will lead
to inconsistency. Therefore, T2 can only be an alternative formulation of T1 if
it explains P1 or an extension if it explains P1 AND P2. If T2 is based a
different axiomatic system A2, then it is uniquely appropriate in that system
just as T1 is uniquely appropriate in A1. Therefore,
Physical Law formulated in Mathematics is uniquely appropriate within the
axiomatic framework of Mathematics used to establish it.
4.
INDISPENSABILITY – CAN YOU DO
PHYSICS WITHOUT MATHEMATICS? SO WHAT?
Mathematical
Realists argue that the fact that Mathematics applies to and explains Physics
lends support Indispensability Argument. Though there are variations of IA, the
popular ones are those attributed to Quine – Putnam and Baker. In Q – P IA, (P1): We ought to have ontological
commitment to all and only the entities that are indispensable to our best
scientific theories. (P2): Mathematical entities are indispensable to our best
scientific theories. (C): We ought to have ontological commitment to
mathematical entities. In Baker’s enhanced IA, (P1): We ought to have ontological commitment to all and only
those entities that play an essential explanatory
role in our best scientific theories. (P2): Mathematical entities play an
essential explanatory role in our
best scientific theories. (C): We ought to have ontological commitment to
mathematical entities. Though P1 has
been the main focus of debate especially by Maddy, in “Science Without
Numbers”, Field offers a significant challenge to P2. In his nominalization
programme, Field agrees that Mathematics is useful, but denies that it is
indispensable, and sets out to demonstrate that it is possible to reformulate
our best scientific theories in a way that doesn’t involve Mathematical Objects.
He does so by providing a ‘nominalist’ reformulation of parts of Newtonian
gravitational theory, a reformulation that replaces terms that purport to refer
to Mathematical Objects with terms that refer to nominalistically acceptable
objects. I don’t think the nominalist programme is essential to establish anti –
realism as there are enough arguments against ontological anti – realism as
seen earlier. History also presents examples of false theories that produce
empirically valid results. Wigner cites the “Free Electron Theory” of Neil Bohr and “Planetary Epicycles of Ptolemy” as examples. This directly
challenges P1 of IA. There are a number of challenges to Nominalist programme
but my fundamental challenge is whether Nominalist Reformulation is
Unreasonably (or Extremely Effective)? We don’t have enough nominalist
reformulations to say that it is an equivalent replacement to Mathematics as we
know and love.
5.
CONCLUSION
Mathematics only
a mind game of manipulating symbols. There are no real Mathematics objects outside
human minds. It is objectively for valid those who agree with the rules of the
game. Mathematics is able to apply to Physics because the latter is also
reducible to a set of symbols in the mental realm. The underlying reality of
Mathematics (mind) and Physics (matter) is indeterminate but produces logically
related symbols in the mind. Mathematics causally explains itself and Physics though
sophistication of the manipulation rules or algorithm makes it difficult to see
it. Mathematics is unreasonably effective because logic operating on symbols,
both are congruent. I call it extremely effective only to communicate that it
is logical not magical though profound. In response to much ignored Wigner’s
second puzzle, I argue that Mathematics is uniquely appropriate to Physics. IA
is a weak argument for realism as scientific theories can be false and
Nominalism incorrectly undermines the position of Mathematics overlooking its
appropriateness. In summary, the applicability of Mathematics to Physics is not
surprising and its effectiveness is not unreasonable but extremely profound. I
must add that Wigner points out the principles of invariance, isolation and
irrelevance that makes Physics possible, which is also true for Mathematics. There
is a profound related question “Why reality seems logical?”!!!!
REFERENCES
1 Colyvan, “An Introduction to the Philosophy of Mathematics” Ch 3.1 p
43
2Colyvan,
“An Introduction to the Philosophy of Mathematics” Ch 3.1 p 47
3Frege, 1903/80 §95,
p. 184 – 190
4The Unreasonable Effectiveness of
Mathematics in Natural Sciences (page 2)
6 Colyvan, “An Introduction to the Philosophy of Mathematics” Ch 6.1 p
114 and 120
7Colyvan,
“An Introduction to the Philosophy of Mathematics” Ch 5
8The Unreasonable Effectiveness of
Mathematics in Natural Sciences (page 4 – 5)
9Quantum Yang Mill’s Theory (page 4)
10 Colyvan, “An Introduction to the Philosophy of Mathematics” Ch 6.1
p 117
11 The Unreasonable Effectiveness of
Mathematics in Natural Sciences (page 9)
12 The Unreasonable Effectiveness of
Mathematics in Natural Sciences (page 9)
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