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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 `∆

A∧B, Γ `∆∧:l1A, Γ `∆

B∧A, 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{x←t}, Γ `∆

(∀x)A, Γ `∆∀:lΓ`∆, A{x←α}

Γ`∆, (∀x)A∀:r

A{x←α}, Γ `∆

(∃x)A, Γ `∆∃:lΓ`∆, A{x←t}

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: I↔Init ↔h(0) = h0∧˙

h(0) = 0

Uniform Gravitation: G↔Gravity ↔(∀t)(f(t) = −mg)

Fall of the Object: F↔Fall ↔(∃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) = −p2gh0∃r

˙

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) = −gα +˙

h(0)),(h(t) = −gα2

2+˙

h(0)α+h(0)),(˙

h(β) = −gβ +˙

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) = −gα +˙

h(0)),(h(t) = −gα2

2+˙

h(0)α+h(0)),(˙

h(β) = −gβ +˙

h(0)),(h(β) = −gβ2

2+˙

h(0)β+h(0)) `gh(α) + ˙

h(α)2

2=gh(β) + ˙

h(β)2

2∀l

(∀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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