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Abstract

Axiomatization of Physics (and science in general) has many drawbacks that are correctly criticized by opposing philosophical views of science. This paper shows that, by giving formal proofs a more prominent role in the formalization, many of the drawbacks can be solved and many of the opposing views are naturally conciliated. Moreover, this approach allows, by means of proof theory, to open new conceptual bridges between the disciplines of Physics and Computer Science. (c) 2011 Elsevier Inc. All rights reserved.
Physics and Proof Theory
Bruno Woltzenlogel Paleo
bruno@logic.at
Institut f¨ur Computersprachen, Vienna University of Technology, Austria
Bruno.WoltzenlogelPaleo@loria.fr
INRIA, LORIA, Nancy, France
Abstract. Axiomatization of Physics (and Science in general) has many
drawbacks that are correctly criticized by opposing philosophical views
of Science. This paper shows that, by giving formal proofs a more promi-
nent role in the formalization, many of the drawbacks can be solved and
many of the opposing views are naturally conciliated. Moreover, this ap-
proach allows, by means of Proof Theory, to open new conceptual bridges
between the disciplines of Physics and Computer Science.
1 Introduction
“Science is built up with facts, as a house is with stones.
But a collection of facts is no more a science
than a heap of stones is a house.”
- Poincar´e
Foundational works on the formalization of Physics typically consider a phys-
ical theory as a collection of facts, i.e. as a set of sentences closed under logical
consequence. However, not as much attention has been given to studying how
these facts are or should be organized or, equivalently, how the physical theory is
or should be structured. Usually, the only structure considered is a distinction of
facts either as axioms or as derivable theorems (i.e. axiomatization). Although
simple, this approach has a few drawbacks.
Firstly, from an epistemological point of view, the mentioned approach suffers
from a logical omniscience problem: although physicists might know the axioms
of their theories, it is certainly not the case that they know all the logical conse-
quences of these axioms, simply because they have limited reasoning resources.
Therefore, the approach of defining a theory as a set of sentences closed un-
der logical consequence fails to capture the notion of theory as perceived by
resource-bounded physicists; it is just an idealized approximation.
Secondly, the selection of which facts should be taken as axioms is arbitrary
and frequently based on subjective criteria such as elegance. For example, there
are axiomatizations of physics that do not rely on the rather natural concepts of
space and time [18]. Should they be considered more elegant, useful or correct?
And finally, there are cases of physical theories, such as Newtonian mechanics
and Lagrangean mechanics, that are considered equivalent to each other accord-
ing to the mentioned approach, because their sets of sentences closed under
logical equivalence are the same, even though they actually differ significantly
in how easily they can be used to solve certain classes of problems.
The second and third drawbacks mentioned above have been main reasons for
criticism on the whole enterprise of formalizing Science [20]. However, they actu-
ally only apply to (unstructured) axiomatization. As a response to the criticism,
there was a rise of semantic approaches, which adopted a more model-theoretic
approach to the formalization of Science [20]. Advances in the sibling discipline
of proof theory, on the other hand, have not been given much attention.
The main goal of this paper is to advocate in favor of a more prominent role
for proofs in the formalization of physics, and consequently, for proof theory in
approaches to Hilbert’s sixth problem [22] and in studies of the foundations of
physics. If a physical theory is considered not as a collection of sentences closed
under logical consequence, but rather as a collection of proofs, the above men-
tioned drawbacks are naturally solved. Non-idealized resource-bounded physi-
cists know only what they have proved so far. Axioms are simply the assumptions
of the proofs contained in the physical theory. And various physical theories can
be objectively compared with respect to the structure of the proofs they contain.
This proposal is in line with current work in the formalization of mathematics,
where mathematical knowledge is formalized as collections of proofs with the
assistance of interactive theorem provers1.
The use of proofs to formalize computations of solutions of physical problems
is exemplified with a simple problem of Newtonian mechanics in Section 3. The
proof calculus used, known as sequent calculus, is briefly explained in Section
2. Finally, Section 4 discusses the benefits and challenges of using proofs in the
formalization of Physics, from philosophical and computational points of view.
2 The Sequent Calculus LKP
The formal proofs in this paper are written in an extension of Gentzen’s sequent
calculus LK [11]. A sequent is a pair Γ`, where Γ(the antecedent) and
(the succedent) are multisets of formulas, with the intuitive intended meaning
that the disjunction of the formulas in is provable assuming the formulas in Γ.
An LK-proof is a (hyper)tree of sequents, such that the leaves are axiom sequents
of the form F`F, where Fis an arbitrary formula, and the (hyper)edges are
instances of the inference rules specified by the calculus. The sequent calculus
LK has inference rules for propositional connectives (e.g. ,,¬and ), as
exemplified below for the connective:
Γ`∆, A Π `Λ, B
Γ, Π `∆, Λ, A B:rA, Γ `
AB, Γ `:l1A, Γ `
BA, Γ `:l2
The following inference rules for quantifiers are also available (with the im-
portant restriction that the :rand :lrules must satisfy the eigenvariable
1Examples of proof assistants are Isabelle (http://www.cl.cam.ac.uk/research/hvg/Isabelle/),
Coq (http://coq.inria.fr/) and Mizar (http://mizar.uwb.edu.pl/).
condition, i.e. the variable αmust occur neither in Γnor in nor in A):
A{xt}, Γ `
(x)A, Γ `:lΓ`∆, A{xα}
Γ`∆, (x)A:r
A{xα}, Γ `
(x)A, Γ `:lΓ`∆, A{xt}
Γ`∆, (x)A:r
Moreover, the sequent calculus LK also provides structural rules such as con-
traction, weakening and, most importantly, the cut rule, which, as discussed in
Section 4, eases the structured formalization of Physics:
Γ`∆, F F, Γ `
Γ`cut
However, the pure sequent calculus LK does not provide any built-in support
for equality handling, arithmetical simplifications, and differentiation and inte-
gration. Therefore, formalizing physics in the pure sequent calculus LK would
be tedious and uncomfortable, since the lack of built-in support would require
adding several additional assumptions to the antecedents of the sequents, which
would render the proofs large, unreadable and difficult to construct. The sequent
calculus LKPaddresses this issue by extending LK with the following rules:
Built-in Support for Equality:
Γ, s =t, A[t]`
Γ, s =t, A[s]`=l
Γ, s =t`∆, A[t]
Γ, s =t`∆, A[s]=r
Γ, s =t, A[s]`
Γ, s =t, A[t]`=l
Γ, s =t`∆, A[s]
Γ, s =t`∆, A[t]=r
where sand tdo not contain variables that are bound in A.
– Built-in Support for Definitions:2They correspond directly to the ex-
tension principle and introduce new predicate and function symbols as ab-
breviations for formulas and terms. Let A[x1, . . . , xk] be an arbitrary for-
mula with free-variables x1, . . . , xkand Pbe a new k-ary predicate symbol
defined by P(x1, . . . , xk)A[x1, . . . , xk]. Let t[x1, . . . , xk] be an arbitrary
term with free-variables x1, . . . , xkand fbe a new k-ary function symbol
defined by f(x1, . . . , xk) = t[x1, . . . , xk]. Then, for arbitrary sequences of
terms t1, . . . , tk, the rules are:
A[t1,...,tk], Γ `
P(t1,...,tk), Γ `dl
Γ`∆, A[t1,...,tk]
Γ`∆, P (t1,...,tk)dr
F[t[t1,...,tk]], Γ `
F[f(t1,...,tk)], Γ `dl
Γ`∆, F [t[t1,...,tk]]
Γ`∆, F [f(t1,...,tk)] dr
2Definition rules have been succesfully used for formalization and analysis of math-
ematical proofs [3]. They are closely related to superdeduction rules [6], which can
provide even more concise, natural and readable formal proofs. However they are
not as simple to describe, and hence definition rules have been used in this paper.
– Built-in Support for Simplification: let t(or t0) be obtainable from t0
(t) by algebraic or arithmetical simplifications3. Then the rules are:
F[t0], Γ `
F[t], Γ `sl
Γ`∆, F [t0]
Γ`∆, F [t]sr
Built-in Support for Integration and Differentation:4let t1(t2) be a
term denoting the integral of the function denoted by t0
1(t0
2) on the interval
(x1, x2). Then the rules are:
F[t1=t2], Γ `
F[t0
1=t0
2], Γ `Rx2
x1:lΓ`∆, F [t0
1=t0
2]
Γ`∆, F [t1=t2]Rx2
x1:r
3 A Simple Example: Energy Conservation as a Cut
To solve problems of physics, certain invariants (such as energy) are frequently
used. This is so because solving problems by using a derived principle (such
as the principle of energy conservation) is usually easier than solving them by
using the most basic physical laws or axioms. This section intends to exemplify
how problem solution can generally be seen from a proof-theoretic perspective
in which the use of derived principles corresponds to an implicit use of the cut
rule. The following simple problem of Newtonian mechanics shall be considered:
An object of mass mis dropped from height h0and with initial
velocity equal to zero. The only force acting on the object is the force of
gravity (with an intensity mg). What is the velocity of the object when
its height is equal to zero?
A typical solution (Solution 1) to this problem uses the principle of energy
conservation, as follows:
1. Let tfbe the time when the object reaches height zero.
2. According to the principle of energy conservation, e(tf) = e(0), i.e. the energy at
tfis equal to the initial energy.
3. Hence, by definition of gravitational potential energy in a uniform gravitational
field and by definition of kinetic energy, mgh(tf) + m˙
h(tf)2
2=mgh(0) + m˙
h(0)2
2.
3It is beyond the scope of this paper to define precisely the allowed simplifications.
This kind of rule is inspired by deduction modulo, whose precise definitions can be
found in [9]. In principle, simplification rules are not necessary, because they can
be simulated by equality rules together with the arithmetical and algebraic axioms
as additional assumptions in the antecedentes of the sequents. However, the goal
of simplification rules (and deduction modulo) is to hide uninteresting computa-
tional details of the underlying theories (e.g. arithmetics), in order to obtain concise
formal proofs that show only interesting information related to the theory under
investigation (e.g. newtonian mechanics).
4Integration and Differentiation Rules have been inspired by emerging idea of inte-
grating computer algebra systems and automated theorem provers.
4. According to the initial conditions, h(0) = h0and ˙
h(0) = 0. Moreover, by assump-
tion, h(tf) = 0. Therefore, m˙
h(tf)2
2=mgh0.
5. Hence, the result is ˙
h(tf) = 2gh0.
Another solution (Solution 2) computes the velocity as a function of time
by integrating the acceleration produced by the gravitational force. Then it
determines the time when the object reaches height zero, and computes the
velocity at that time. The details are shown below:
1. According to Newton’s second law of motion, f(t) = m¨
h(t) at any time t. Moreover,
the uniform gravitational field produces a force f(t) = mg. Hence, ¨
h(t) = g.
2. By integration, ˙
h(t) = gt +˙
h(0).
3. According to the initial conditions, ˙
h(0) = 0, and hence ˙
h(t) = gt.
4. By integration again, h(t) = gt2
2+h(0).
5. According to the initial conditions, h(0) = h0, and hence h(t) = gt2
2+h0.
6. For h(tf) = 0 to hold, it must be the case that tf=q2h0
g.
7. Hence ˙
h(tf) = gq2h0
g, which can be simplified to ˙
h(tf) = 2gh0.
Solution 2 is simpler in the sense that it uses only the basic physical laws of
motion (here assumed to be Newton’s laws of motion) and of uniform gravita-
tional fields. Solution 1, on the other hand, assumes that energy is conserved,
without actually proving it from Newton’s basic laws.
In order to view problem solving from a proof theoretic perspective, it is
necessary to formalize problem solving as theorem proving. In the example above,
the problem can be stated as the following theorem to be proved:
(t0)(h(t0)=0(v)˙
h(t0) = v)
Solving the given problem then consists of finding a proof of the theorem
above such that vis instantiated by a ground term. Interestingly, formalizing
the problem as a theorem to be proved enforces the explicit mention of the hidden
assumption that the height eventually becomes zero; otherwise the variable t0
would be free and the theorem would be open.
Traditionally, works of axiomatization have formalized physical laws as ax-
ioms that are supposed to be used as assumptions in proofs [20]. In a more mod-
ern proof-theoretical approach, however, definition rules often provide a more
convenient alternative. The axioms corresponding to certain physical laws can
be seen as defining new symbols. This is the case, for example, of Newton’s second
law, which states that force equals mass times acceleration (f(t) = m¨
h(t)). It can
be seen as defining the function symbol f. Similarly, the equation for energy of a
single object in a uniform newtonian gravitational field (e(t) = mgh(t) +m˙
h(t)2
2)
can be seen as defining the function symbol e.For convenience, the defined pred-
icate symbols below are also used in the following formal proofs:
Initial Conditions: IInit h(0) = h0˙
h(0) = 0
Uniform Gravitation: GGravity (t)(f(t) = mg)
Fall of the Object: FFall (t)h(t) = 0
Energy Conservation: EC EnergyConservation (ti)(tj)e(ti) = e(tj)
Solution 1 can be easily formalized as the proof ϕ1below (where ϕ0
1is a
subproof consisting of the single axiom sequent h(tf)=0`h(tf) = 0):
ϕ0
1
˙
h(tf) = p2gh0`˙
h(tf) = p2gh0r
˙
h(tf) = p2gh0`(v)˙
h(tf) = v
sl
mg0 + m
˙
h(tf)2
2=mgh0+m02
2`(v)˙
h(tf) = v
wl
h(tf)=0, h(0) = h0,˙
h(0) = 0, mg0 + m
˙
h(tf)2
2=mgh0+m02
2`(v)˙
h(tf) = v
=l
h(tf)=0, h(0) = h0,˙
h(0) = 0, mgh(tf) + m
˙
h(tf)2
2=mgh(0) + m˙
h(0)2
2`(v)˙
h(tf) = v
dl
h(tf)=0, h(0) = h0,˙
h(0) = 0, e(tf) = e(0) `(v)˙
h(tf) = v
l
h(tf)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`(v)˙
h(tf) = v
l
h(tf)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`(v)˙
h(tf) = v
r
h(tf)=0, h(tf)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`h(tf)=0(v)˙
h(tf) = v
cl
h(tf)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`h(tf)=0(v)˙
h(tf) = v
r
h(tf)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`(t0)(h(t0)=0(v)˙
h(t0) = v)
l
(t)h(t)=0, h(0) = h0,˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`(t0)(h(t0)=0(v)˙
h(t0) = v)
l
(t)h(t)=0, h(0) = h0˙
h(0) = 0,(ti)(tj)e(ti) = e(tj)`(t0)(h(t0)=0(v)˙
h(t0) = v)
dl
Fall,Init ,EnergyConservation `(t0)(h(t0)=0(v)˙
h(t0) = v)
Solution 2 can be formalized as the following proof ϕ2:
h r2h0
g!= 0 `h r2h0
g!= 0
wl
h(0) = h0, h r2h0
g!= 0 `h r2h0
g!= 0
˙
h r2h0
g!=p2gh0`˙
h r2h0
g!=p2gh0
r
˙
h r2h0
g!=p2gh0`(v)˙
h r2h0
g!=v
sl
˙
h r2h0
g!=gr2h0
g`(v)˙
h r2h0
g!=v
l
(t)( ˙
h(t) = gt)`(v)˙
h r2h0
g!=v
wl
˙
h(0) = 0,(t)( ˙
h(t) = gt)`(v)˙
h r2h0
g!=v
r
h(0) = h0,˙
h(0) = 0, h r2h0
g!= 0,(t)( ˙
h(t) = gt)`h r2h0
g!= 0 (v)˙
h r2h0
g!=v
r
h(0) = h0,˙
h(0) = 0, h r2h0
g!= 0,(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)
sl
h(0) = h0,˙
h(0) = 0, h r2h0
g!=g r2h0
g!2
2+h0,(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)
l
h(0) = h0,˙
h(0) = 0,(t)(h(t) = gt2
2+h0),(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)=l
h(0) = h0,˙
h(0) = 0,(t)(h(t) = gt2
2+h(0)),(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)
Rl
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt),(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)cl
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt)`(t0)(h(t0)=0(v)˙
h(t0) = v)sl
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt + 0) `(t0)(h(t0)=0(v)˙
h(t0) = v)=l
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt +˙
h(0)) `(t0)(h(t0)=0(v)˙
h(t0) = v)Rl
h(0) = h0,˙
h(0) = 0,(t)(¨
h(t) = g)`(t0)(h(t0)=0(v)˙
h(t0) = v)sl
h(0) = h0,˙
h(0) = 0,(t)(m¨
h(t) = mg)`(t0)(h(t0)=0(v)˙
h(t0) = v)dl
h(0) = h0,˙
h(0) = 0,(t)(f(t) = mg)`(t0)(h(t0)=0(v)˙
h(t0) = v)l
h(0) = h0˙
h(0) = 0,(t)(f(t) = mg)`(t0)(h(t0)=0(v)˙
h(t0) = v)
dl
Init,Gravity `(t0)(h(t0)=0(v)˙
h(t0) = v)
As expected ϕ1is not only smaller than ϕ2, but also simpler in the sense that
it does not use integration. Furthermore, while in ϕ2the time when the object
hits the floor has to be computed explicitly (i.e. t0is instantiated to a ground
term), in ϕ1this is not so (i.e. t0is instantiated to a variable).
Solution 1 implicitly uses cuts, because EnergyConservation and Fall are not
considered to be basic laws of physics. In principle, ϕ1must be composed with a
proof ϕEof EnergyConservation and a proof ϕFof Fall. This is done with two
cuts, as shown in the following proof ϕ:
ϕF
Init,Gravity `Fall
ϕE
Gravity `EC
ϕP
Init,Fall ,EC `(t0)(h(t0)=0(v)˙
h(t0) = v)cut
Init,Gravity ,Fall `(t0)(h(t0)=0(v)˙
h(t0) = v)cut
Init,Init ,Gravity,Gravity `(t0)(h(t0)=0(v)˙
h(t0) = v)cl
Init,Gravity `(t0)(h(t0)=0(v)˙
h(t0) = v)
Where ϕFis the proof below, proving that the object will eventually fall to
height zero under the gravitational field and the initial conditions specified in
the description of the problem:
q2h0
g= 0 `hq2h0
g= 0
r
hq2h0
g= 0 `(t0)h(t0) = 0
sl
hq2h0
g=g r2h0
g!2
2+h0`(t0)h(t0) = 0
l
(t)(h(t) = gt2
2+h0)`(t0)h(t0) = 0 wl
h(0) = h0,(t)(h(t) = gt2
2+h0)`(t0)h(t0) = 0 =l
h(0) = h0,(t)(h(t) = gt2
2+h(0)) `(t0)h(t0) = 0 Rl
h(0) = h0,(t)( ˙
h(t) = gt)`(t0)h(t0) = 0 wl
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt + 0) `(t0)h(t0) = 0 sl
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt + 0) `(t0)h(t0) = 0 =l
h(0) = h0,˙
h(0) = 0,(t)( ˙
h(t) = gt +˙
h(0)) `(t0)h(t0) = 0 Rl
h(0) = h0,˙
h(0) = 0,(t)(¨
h(t) = g)`(t0)h(t0) = 0 sl
h(0) = h0,˙
h(0) = 0,(t)(m¨
h(t) = mg)`(t0)h(t0) = 0 dl
h(0) = h0,˙
h(0) = 0,(t)(f(t) = mg)`(t0)h(t0) = 0 l
h(0) = h0˙
h(0) = 0,(t)(f(t) = mg)`(t0)h(t0) = 0 d
Init,Gravity `Fall
And ϕEis the proof that energy is conserved in a uniform gravitational field:
`gh(0) + ˙
h(0)2
2=gh(0) + ˙
h(0)2
2wl
(˙
h(α) = +˙
h(0)),(h(t) = gα2
2+˙
h(0)α+h(0)),(˙
h(β) = +˙
h(0)),(h(β) = gβ2
2+˙
h(0)β+h(0)) `gh(0) + ˙
h(0)2
2=gh(0) + ˙
h(0)2
2=
r, s
r
(˙
h(α) = +˙
h(0)),(h(t) = gα2
2+˙
h(0)α+h(0)),(˙
h(β) = +˙
h(0)),(h(β) = gβ2
2+˙
h(0)β+h(0)) `gh(α) + ˙
h(α)2
2=gh(β) + ˙
h(β)2
2l
(t)( ˙
h(t) = gt +˙
h(0)),(t)(h(t) = gt2
2+˙
h(0)t+h(0)),(t)( ˙
h(t) = gt +˙
h(0)),(t)(h(t) = gt2
2+˙
h(0)t+h(0)) `gh(α) + ˙
h(α)2
2=gh(β) + ˙
h(β)2
2cl
(t)( ˙
h(t) = gt +˙
h(0)),(t)(h(t) = gt2
2+˙
h(0)t+h(0)) `gh(α) + ˙
h(α)2
2=gh(β) + ˙
h(β)2
2cl
(t)( ˙
h(t) = gt +˙
h(0)) `gh(α) + ˙
h(α)2
2=gh(β) + ˙
h(β)2
2Rl
(t)(¨
h(t) = g)`gh(α) + ˙
h(α)2
2=gh(β) + ˙
h(β)2
2s
(t)(m¨
h(t) = mg)`mgh(α) + m˙
h(α)2
2=mgh(β) + m˙
h(β)2
2dr
(t)(m¨
h(t) = mg)`e(α) = e(β)dl
(t)(f(t) = mg)`e(α) = e(β)
r
(t)(f(t) = mg)`e(α) = e(β)
r
(t)(f(t) = mg)`(ti)(tj)e(ti) = e(tj)
d
Gravity `EnergyConservation
4 Benefits and Challenges of a Proof-Theoretical
Approach to the Formalization of Physics
The following subsections are devoted to discussing what proof theory has to
offer to the formalization of Physics, with emphasis on computational and philo-
sophical aspects.
4.1 Cut-Introduction
The example discussed in the previous section illustrates that an essential task
of theoretical science is to invent or discover important concepts that are useful
to solve problems, such as the principle of energy conservation in newtonian me-
chanics. Nevertheless, in a traditional axiomatization approach, such principles
have no prominent role, because they are merely theorems derivable from the
axioms. In a more proof-theoretic approach, on the other hand, proofs allow a
structured formalization of the scientific knowledge, where important principles
like energy conservation appear prominently formalized as active formulas in
cut inferences, as shown in the formal proof ϕof Section 3. Indeed, reduction-
ism in Science can generally be captured by the proof-theretical notion of cut.
Consequently, a significant part of the usual scientific activity can be formally
described as cut-introduction.
Cut-introduction also leads to the compression of proofs. Although the gen-
eral problem of finding the shortest proofs by means of cut-introduction is unde-
cidable [5], there are a few preliminary algorithms that introduce simple cuts [15,
10, 24], and it has been shown that some techniques of machine learning, such as
decision tree learning, can be seen as cut-introduction techniques from a proof-
theoretical point of view [23]. Therefore, a potential benefit of using proofs to
formalize Physics is the possibility of applying cut-introduction techniques in or-
der to automatically discover useful physical concepts. However, it must be noted
that current cut-introduction techniques are still not sophisticated enough to be
applied to formalized proofs of Physics.
4.2 Cut-Elimination
The problem of eliminating cuts from proofs is much easier than the problem of
introducing cuts and has been much more deeply investigated [11, 4]. By using
cut-elimination algorithms, it might be possible to automatically transform a
solution that uses a derived principle (i.e. a cut) such as energy conservation
(e.g. Solution 1 in Section 3) into a solution that uses only the basic laws of
a theory (e.g. Solution 2 in Section 3). This is advantageous in certain cases,
for in a cut-free proof it is easy, via Gentzen’s Midsequent Theorem [11] or
more general Herbrand sequent extraction algorithms [16], to extract a Herbrand
disjunction [12] that contains instances of the quantified variables of the problem.
For example, in the cut-free proof of Solution 2, the existentially quantified
variable for the time when the object reaches height zero is instantiated by a
ground term that denotes exactly when this happens. In the proof with cuts that
formalizes Solution 1, on the other hand, it is instantiated by an eigenvariable,
and hence the time when the object reaches height zero is not known. Therefore,
cut-elimination could in principle be used as an algorithm that instantiates the
variables of a problem that were left unsolved. However, even though this idea
has been succesfully used in mathematics [14], the challenge in the case of Physics
is to make cut-elimination algorithms work with high-level calculi such as LKP.
4.3 Logic Programming
The idea of formalizing a problem as a theorem and in such a way that its solution
is in the instances used for the quantified variables in the proof is the fundamen-
tal principle behind the logic programming paradigm of computation, of which
Prolog [19] is the most prominent language. Therefore, the proof-theoretical ap-
proach to the formalization of Physics brings a new paradigm of computation
that might be the subject of studies from the point of view of Physics itself, as
imperative computation, which is modeled by Turing machines, has been.
4.4 Functional Programming and the Curry-Howard Isomorphism
The Curry-Howard isomorphism [8] states that there is a correspondence be-
tween proofs of the implicational fragment of intuitionistic logic and lambda
terms. A proof is essentially a functional program. Cut-elimination corresponds
to beta-reduction, which is the execution of the program. Cut-introduction cor-
responds to structuring of the program and possibly to code reuse. By extrap-
olating this isomorphism, theories of Physics formalized as collections of proofs
can be seen as collections of programs. This kind of computation, which is im-
plicit in the formalization of Physics, is yet another link between Physics and
computation that might be the target of future work.
4.5 Instrumentalism: Truth versus Usefulness
From an instrumental viewpoint, “the most important function of a theory is
not to organize or assert statements that are true or false but to furnish material
principles of inference that may be used in inferring one set of facts from an-
other”. This idea is supported by the proof-theoretical approach described here,
as shown in the formal proof ϕ2in Section 3, where Newton’s law of motion was
not merely a statement; it was used as a principle of inference, in the form of a
definition inference rule. Instrumentalism also judges theories by how useful they
are in solving problems. The proof-theoretical approach naturally embraces this
criterium of usefulness, since solutions to problems can be formalized as proofs,
as shown by ϕ1and ϕ2. And as the commitment to truth is not given up, it
conciliates two opposing positions in the philosophy of science.
4.6 The Evolution of Theories
Another philosophical viewpoint that opposes axiomatization is that of Weltan-
schauungen analyses, according to which science ought to be viewed as “an on-
going social enterprise [and] epistemic understanding of scientific theories could
only be had by seeing the dynamics of theory development” [20]. “An ultimately
meaningful answer to the question ‘what is a scientific theory?’ cannot be given
in terms of the kinds of concepts considered earlier [axiomatization and seman-
tics]. An adequate and complete answer can be given only in terms of an explicit
and detailed consideration of both the producers and consumers of the theory.”
[21]. Proof theory conciliates formalization with this philosophical viewpoint in
the following way: by defining scientific theories as collections of proofs, they can
evolve by the addition of new proofs, and Kuhn’s major paradigm shifts can be
seen as major proof transformations (e.g. cut-elimination, cut-introduction and
addition of new definitions).
4.7 Algorithmic Information Theory
Algorithmic Information Theory (AIT) sees scientific theories as data com-
pressed in the form of programs. It provides a very simple, elegant and general
criterium to judge and compare theories: the smaller the program, the better
the theory. However, the proponents of AIT are currently making an unfortu-
nate choice of how to encode their data, and this causes the limitations of their
approach. Diagrams in [7] suggest that theories/programs should correspond to
axioms, and the execution of the program by a computer, regarded as an auto-
mated theorem prover, should output empirical data in the form of theorems.
Therefore, they essentially adhere to the traditional Hilbert-style axiomatization
approach, and hence they suffer the same drawbacks, which are nicely explained
from a computational point of view in [7]. Two of them can be summarized
as follows: in current AIT, computation time is ignored, because only program
size matters; and the theory/program’s language is static, implying that new
concepts can never emerge and the theory can never evolve.
Fortunately, proof theory can rescue AIT as well, and even provide further
insight. The idea is that AIT’s principle of program-size minimality should be
applied not to axioms (artificially encoded as programs) but rather to the proofs
that formalize a scientific theory. From a conceptual point of view, it is clear that
proof theory and AIT fit perfectly together, because proofs are already programs
according to the (extrapolated) Curry-Howard isomorphism. The computation
time that was previously ignored now appears explicitly as the length of proofs
[17] and theories can naturally evolve by the addition and transformation of
proofs in the collection, with new concepts emerging by the introduction of cuts
and definition inferences.
Another indication that AIT and proof theory fit well together is the natural
relation between cut-introduction and kolmogorov complexity [13]. The Kol-
mogorov complexity C(ψ) of a proof ψcan be defined as the size of the shortest
proof ψ0that can be obtained by cut-introduction from ψ(and, conversely, such
that ψcan be reconstructed from ψ0by cut-elimination).
5 Conclusions
“It is unheard of to find a substantive example of a theory actually worked
out as a logical calculus in the writings of most philosophers of science. Much
handwaving is indulged in to demonstrate that this [.. . ] is simple in principle
and only a matter of tedious detail, but concrete evidence is seldom given.”
[21]. In Section 3, an example of problem solution in Newtonian mechanics has
been successfully worked out in a sequent calculus extended with sophisticated
simplification, integration and definition rules, inspired by recent advances in
Proof Theory. These extensions are the key to the small size and significantly
reduced amount of tedious detail in the obtained formal proofs.
Section 4 showed that this proof-theoretical approach successfully conciliates
and unifies various philosophical views of Science, such as formalism, instrumen-
talism and Weltanschauungen analyses. The essence of these achievements lies
in seeing scientific theories not just as collections of facts, as assumed by tradi-
tional axiomatization. Scientific theories ought to be formalized as collections of
proofs. The structure of scientific knowledge can be nicely formalized with cuts,
and much of the scientific activity can be formally described as proof generation
or proof transformation. The task of organizing knowledge, for example, can be
formally described as cut-introduction.
Moreover, cut-introduction potentially compresses proofs, which can also be
seen as programs according to the (extrapolated) Curry-Howard isomorphism.
This indicates a tight relation between cut-introduction and Kolmogorov com-
plexity, and thus the use of proofs clarifies, conceptually improves and solves
some limitations of the ideas of algorithmic information theory with respect to
the formalization of Science.
The proof-theoretical approach advocated here should be seen not as com-
peting against existing axiomatic and semantical approaches, but rather as com-
plementing them by enriching their formalizations with structure.
Future work should concentrate on applying these proof-theoretical ideas to
complement the formalization of more interesting physical theories, such as Rel-
ativity (e.g. [2]) and Quantum Mechanics (e.g. [1]); on improving proof assistants
and proof-theoretical techniques, such as cut-elimination and cut-introduction,
in order to support logical calculi at least as sophisticated as LKP; and on in-
vestigating the new links between Physics and Computation that are opened by
Proof Theory.
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... Furthermore, in the formalization of mathematical proofs, lemmas correspond to cuts, and hence the automatic introduction of cuts is, in a formal level, the automatic discovery of lemmas that are potentially useful for structuring mathematical knowledge. Naturally, the use of cut-introduction techniques could in principle also be applied to the structuring of knowledge in other fields of Science, as argued in [18, 19]. This paper presents a new method for the introduction of atomic cuts: CIRes. ...
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Software technology that can be used to validate the logical correctness of mathematical proofs has attained a high degree of power and sophistication; extremely difficult and complex mathematical theorems have been verified. This paper discusses the prospects of doing something comparable for elementary physics: what it would mean, the challenges that would have to be overcome; and the potential impact, both practical and theoretical. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
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Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzen’s sequent calculus LK for classical first-order logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES.
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