<figure><img src="image_1.png" /></figure>

HAL
open science

# Explanation: from ethics to logic

Gilles Dowek

► To cite this version:

Gilles Dowek. Explanation: from ethics to logic. Annals of the Japan Association for Philosophy of Science, 2023, 32, pp.16. hal-04055141v5

HAL Id: hal-04055141

https://inria.hal.science/hal-04055141v5

Submitted on 16 Jan 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_1.png" /></figure>

HAL
open science

# Explanation: from ethics to logic

Gilles Dowek

► To cite this version:

Gilles Dowek. Explanation: from ethics to logic. Annals of the Japan Association for Philosophy of Science, 2023, 32, pp.16. hal-04055141v5

HAL Id: hal-04055141

https://inria.hal.science/hal-04055141v5

Submitted on 16 Jan 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Explanation: from ethics to logic

Gilles Dowek*

## Abstract

When a decision, such as the approval or denial of a bank loan, is delegated to a computer, an explanation of that decision ought to be given with it. This ethical need to explain the decisions leads to the search for a formal definition of the notion of explanation. This question meets older questions in logic regarding the explanatory nature of proof.

Keywords: explanation, ethics, proof, cut, generalization

## 1 Explanation: an ethical need

When a person makes a decision regarding another person, for example when a clerk refuses a bank loan to a customer, she ought to explain her decision. Providing such an explanation realizes several values: transparency, equality, agency, dignity... Transparency, because the customer wants to know the rules according to which her application has been rejected. Equality, because she wants to make sure these rules are the same for everyone. Agency, because providing such an explanation enables the customer to improve her application. Dignity, because providing an explanation sets up the customer as a rational being, at the same level as the clerk.

If the clerk delegates this decision to a computer, such an explanation of the decision must be generated together with the decision itself. Automatizing decision thus requires a formal definition of the notion of explanation, while the interaction between persons only required an informal one: a common agreement.

<footnote>*Inria, ENS-Paris Saclay, and CNPEN, gilles.dowek@ens-paris-saclay.fr.</footnote>
# Explanation: from ethics to logic

Gilles Dowek*

## Abstract

When a decision, such as the approval or denial of a bank loan, is delegated to a computer, an explanation of that decision ought to be given with it. This ethical need to explain the decisions leads to the search for a formal definition of the notion of explanation. This question meets older questions in logic regarding the explanatory nature of proof.

Keywords: explanation, ethics, proof, cut, generalization

## 1 Explanation: an ethical need

When a person makes a decision regarding another person, for example when a clerk refuses a bank loan to a customer, she ought to explain her decision. Providing such an explanation realizes several values: transparency, equality, agency, dignity... Transparency, because the customer wants to know the rules according to which her application has been rejected. Equality, because she wants to make sure these rules are the same for everyone. Agency, because providing such an explanation enables the customer to improve her application. Dignity, because providing an explanation sets up the customer as a rational being, at the same level as the clerk.

If the clerk delegates this decision to a computer, such an explanation of the decision must be generated together with the decision itself. Automatizing decision thus requires a formal definition of the notion of explanation, while the interaction between persons only required an informal one: a common agreement.

<footnote>*Inria, ENS-Paris Saclay, and CNPEN, gilles.dowek@ens-paris-saclay.fr.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: If a point $M$ is on the perpendicular bisectors the segments $AB$ and $BC$, then $d(M, A) = d(M, B)$ and $d(M, B) = d(M, C)$, hence $d(M, A) = d(M, C)$ and $M$ is also on the perpendicular bisector of $AC$.</figcaption></figure>

# 2 Proof as an explanation

A first attempt is to define an explanation of a statement as a logical proof of this statement. For instance the proof, Figure 1, that the three perpendicular bisectors of the sides of a triangle meet at one point, both shows that these bisectors meet at one point, and also explains why.

The statement “the perpendicular bisectors of the sides of the triangle meet at one point” is the thing that is explained and the proof of this statement is the thing that explains it.

# 3 Four proofs that are not explanations

But, in a discussion, more epistemological than ethical [3], we have already pointed that some proofs do not seem to explain anything.

## 3.1 The four color theorem

A first example is the proof of the four color theorem expressing that every map is four colorable (Figure 2). This proof, constructed in 1976 by Appel, Haken, and Koch [1, 2], reduces the infinitude of all possible maps to 1482 and checks, one by one, that these 1482 maps verify some property, called *reducibility*. This long case study proves that the 1482 maps are all reducible, but it does not seem to explain why. Worse: this case study rather calls for
<figure><img src="image_3.png" /><figcaption>Figure 1: If a point $M$ is on the perpendicular bisectors the segments $AB$ and $BC$, then $d(M, A) = d(M, B)$ and $d(M, B) = d(M, C)$, hence $d(M, A) = d(M, C)$ and $M$ is also on the perpendicular bisector of $AC$.</figcaption></figure>

# 2 Proof as an explanation

A first attempt is to define an explanation of a statement as a logical proof of this statement. For instance the proof, Figure 1, that the three perpendicular bisectors of the sides of a triangle meet at one point, both shows that these bisectors meet at one point, and also explains why.

The statement “the perpendicular bisectors of the sides of the triangle meet at one point” is the thing that is explained and the proof of this statement is the thing that explains it.

# 3 Four proofs that are not explanations

But, in a discussion, more epistemological than ethical [3], we have already pointed that some proofs do not seem to explain anything.

## 3.1 The four color theorem

A first example is the proof of the four color theorem expressing that every map is four colorable (Figure 2). This proof, constructed in 1976 by Appel, Haken, and Koch [1, 2], reduces the infinitude of all possible maps to 1482 and checks, one by one, that these 1482 maps verify some property, called *reducibility*. This long case study proves that the 1482 maps are all reducible, but it does not seem to explain why. Worse: this case study rather calls for
<figure><img src="image_4.png" /><figcaption>Figure 2: Four colors are sufficient to color any map, in such a way that each country is of a color different from that of its neighbors.</figcaption></figure>

an explanation, than it provides one, as if we drew 1482 triangles and checked
that the perpendicular bisectors of the sides of each triangle met at one point,
we would rather think that such a coincidence needs to be explained, than
that it is, itself, an explanation.

## 3.2 The weather forecast

Another example of a proof that does not seem to provide any explanation
is the computation of the weather forecast. For example, if this computation
forecasts a temperature of 19°C in Tokyo tomorrow (Figure 3), then we can
view this computation as a proof of the statement “the temperature in Tokyo
tomorrow will be 19°C”.

Of course, unlike the judgement that the perpendicular bisectors of the
sides of the triangle meet at one point, that is analytic, the judgement that
the temperature in Tokyo tomorrow will be 19°C is synthetic, thus the state-
ment “the temperature in Tokyo tomorrow will be 19°C” cannot be proved.
So, this computation is a proof that the temperature in Tokyo tomorrow will
be 19°C, not in the real world, but in a fictive one, defined by the axioms
of the implemented numerical analysis model. That this fictive world ap-
proximates the real world well enough is a falsifiable statement, that can be
temporarily supported by an empirical comparison of the temperature series
in the real world and in the fictive one.

This proof is not explanatory and the reason why it is not seems to be
that the numerical analysis computation leading to this forecast is very long,
even when using a massively parallel computer. Each step of the computation
<figure><img src="image_4.png" /><figcaption>Figure 2: Four colors are sufficient to color any map, in such a way that each country is of a color different from that of its neighbors.</figcaption></figure>

an explanation, than it provides one, as if we drew 1482 triangles and checked
that the perpendicular bisectors of the sides of each triangle met at one point,
we would rather think that such a coincidence needs to be explained, than
that it is, itself, an explanation.

## 3.2 The weather forecast

Another example of a proof that does not seem to provide any explanation
is the computation of the weather forecast. For example, if this computation
forecasts a temperature of 19°C in Tokyo tomorrow (Figure 3), then we can
view this computation as a proof of the statement “the temperature in Tokyo
tomorrow will be 19°C”.

Of course, unlike the judgement that the perpendicular bisectors of the
sides of the triangle meet at one point, that is analytic, the judgement that
the temperature in Tokyo tomorrow will be 19°C is synthetic, thus the state-
ment “the temperature in Tokyo tomorrow will be 19°C” cannot be proved.
So, this computation is a proof that the temperature in Tokyo tomorrow will
be 19°C, not in the real world, but in a fictive one, defined by the axioms
of the implemented numerical analysis model. That this fictive world ap-
proximates the real world well enough is a falsifiable statement, that can be
temporarily supported by an empirical comparison of the temperature series
in the real world and in the fictive one.

This proof is not explanatory and the reason why it is not seems to be
that the numerical analysis computation leading to this forecast is very long,
even when using a massively parallel computer. Each step of the computation
<figure><img src="image_5.png" /><figcaption>Figure 3: Why 19°C and not 18°C?</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Cats and dogs</figcaption></figure>

seems to contain a bit of explanation, that we cannot aggregate, when all
these steps are put together.

3.3 Data-centric algorithms

Another example of a proof that does not seem to provide any explanation is
the use of data-centric algorithms, for instance machine learning algorithms.

On Figure 4, the points represent pictures in a database. The red points represent cat pictures and the blue ones dog pictures. The new point, in white, is, most likely, a dog picture as its average distance to dog pictures is smaller than that to cat pictures. But this proof of the statement “The new picture most likely represents a dog” does not explain why this picture represents, most likely, a dog, as would a proof analyzing the size of the ears or the shape of the muzzle of the animal.
<figure><img src="image_5.png" /><figcaption>Figure 3: Why 19°C and not 18°C?</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Cats and dogs</figcaption></figure>

seems to contain a bit of explanation, that we cannot aggregate, when all
these steps are put together.

3.3 Data-centric algorithms

Another example of a proof that does not seem to provide any explanation is
the use of data-centric algorithms, for instance machine learning algorithms.

On Figure 4, the points represent pictures in a database. The red points represent cat pictures and the blue ones dog pictures. The new point, in white, is, most likely, a dog picture as its average distance to dog pictures is smaller than that to cat pictures. But this proof of the statement “The new picture most likely represents a dog” does not explain why this picture represents, most likely, a dog, as would a proof analyzing the size of the ears or the shape of the muzzle of the animal.
$$ 
\begin{array}{r}
7678 \\
3706 \\
\hline
46068 \\
00000 \\
53746 \\
23034 \\
\hline
28454668
\end{array}
 $$

<caption>Figure 5: A multiplication</caption>

In this case, the reason why the proof is not explanatory seems to be the large amount of data used by the algorithm. Each piece of data seems to contain a bit of explanation, that we cannot aggregate, when all these data are put together.

## 3.4 The multiplication of arbitrary numbers

A final example of a proof that does not seem to provide any explanation is the multiplication on Figure 5, of two randomly chosen numbers. This multiplication proves that $7678 \times 3706 = 28454668$ but does not explain why this is the case.

A similar example is that of an algorithm that computes a pseudo-random number: it would be non sense to attempt to explain why this number is 28454668.

These four examples: the four color theorem, the weather forecast, the data-centric algorithms, and the multiplication, are four proofs that do not seem to provide any explanation of the proved statement. In some cases, it is even difficult to imagine what an explanation would look like: if we pick two arbitrary numbers and multiply them, what is there to explain?

# 4 Another multiplication

Unlike in that of Figure 5, there seems to be something to explain in the multiplication of Figure 6: one can observe some regularity both in the first number to be multiplied: 12345679, although the 8 seems to be missing, and in the result 444444444. Thus, there seems to be something to explain: we
$$ 
\begin{array}{r}
7678 \\
3706 \\
\hline
46068 \\
00000 \\
53746 \\
23034 \\
\hline
28454668
\end{array}
 $$

<caption>Figure 5: A multiplication</caption>

In this case, the reason why the proof is not explanatory seems to be the large amount of data used by the algorithm. Each piece of data seems to contain a bit of explanation, that we cannot aggregate, when all these data are put together.

## 3.4 The multiplication of arbitrary numbers

A final example of a proof that does not seem to provide any explanation is the multiplication on Figure 5, of two randomly chosen numbers. This multiplication proves that $7678 \times 3706 = 28454668$ but does not explain why this is the case.

A similar example is that of an algorithm that computes a pseudo-random number: it would be non sense to attempt to explain why this number is 28454668.

These four examples: the four color theorem, the weather forecast, the data-centric algorithms, and the multiplication, are four proofs that do not seem to provide any explanation of the proved statement. In some cases, it is even difficult to imagine what an explanation would look like: if we pick two arbitrary numbers and multiply them, what is there to explain?

# 4 Another multiplication

Unlike in that of Figure 5, there seems to be something to explain in the multiplication of Figure 6: one can observe some regularity both in the first number to be multiplied: 12345679, although the 8 seems to be missing, and in the result 444444444. Thus, there seems to be something to explain: we
$$ \begin{array}{r|r} 12345679 & \\ \cline{2-2} 36 & \\ \cline{2-2} 74074074 & \\ \cline{2-2} 37037037 & \\ \cline{2-2} 444444444 & \\ \end{array} $$

Figure 6: If there were a 5 in the result, it would probably be a mistake.

can seek for the reason why the result only contains the digit 4.

## 4.1 A first bit of explanation

Of course, a mathematically mature eye quickly notices that the second number to be multiplied, 36, has something to do with 4: it is $9 \times 4$. So, the proved statement rephrases

$$ 12345679 \times 9 \times 4 = 111111111 \times 4 $$

and this statement has another, more explanatory, proof than the multiplication of Figure 6, obtained by first proving the lemma

$$ 12345679 \times 9 = 11111111 $$

and then multiplying both sides by 4.

But, if both sides of this statement can be multiplied by 4, they can also be multiplied by any digit, for instance 7, yielding the statement

$$ 12345679 \times 63 = 777777777 $$

Both theorems are consequences of a more general one

$$ \forall n \in [1, 9] 12345679 \times 9 \times n = 11111111 \times n $$

and the explanatory nature of the second proof of the statement seems to be a consequence of the fact that it proceeds by proving this general result and then specializing it to $n = 4$.

This general result is the basis of a magic trick due to Lewis Carroll [4, p.78]: at a children's party, Lewis Carroll asked a child to name her favorite digit, in his head he multiplied it by 9 and asked the child to multiply
$$ \begin{array}{r|r} 12345679 & \\ \cline{2-2} 36 & \\ \cline{2-2} 74074074 & \\ \cline{2-2} 37037037 & \\ \cline{2-2} 444444444 & \\ \end{array} $$

Figure 6: If there were a 5 in the result, it would probably be a mistake.

can seek for the reason why the result only contains the digit 4.

## 4.1 A first bit of explanation

Of course, a mathematically mature eye quickly notices that the second number to be multiplied, 36, has something to do with 4: it is $9 \times 4$. So, the proved statement rephrases

$$ 12345679 \times 9 \times 4 = 111111111 \times 4 $$

and this statement has another, more explanatory, proof than the multiplication of Figure 6, obtained by first proving the lemma

$$ 12345679 \times 9 = 11111111 $$

and then multiplying both sides by 4.

But, if both sides of this statement can be multiplied by 4, they can also be multiplied by any digit, for instance 7, yielding the statement

$$ 12345679 \times 63 = 777777777 $$

Both theorems are consequences of a more general one

$$ \forall n \in [1, 9] 12345679 \times 9 \times n = 111111111 \times n $$

and the explanatory nature of the second proof of the statement seems to be a consequence of the fact that it proceeds by proving this general result and then specializing it to $n = 4$.

This general result is the basis of a magic trick due to Lewis Carroll [4, p.78]: at a children's party, Lewis Carroll asked a child to name her favorite digit, in his head he multiplied it by 9 and asked the child to multiply
<figure><img src="image_7.png" /><figcaption>Figure 7: The regularity is also in the partial remainders: 1, 2, 3...</figcaption></figure>

12345679 by the product. She then got a raw of nine repetitions of the named digit.

But this explanation is, of course, partial, as the statement

$$12345679 \times 9 = 11111111$$

still remains to be explained.

## 4.2 A deeper explanation

Instead of explaining the statement

$$12345679 \times 9 = 11111111$$

we can attempt to explain the equivalent one

$$11111111/9 = 12345679$$

If we perform the division of 11111111 by 9 (Figure 7), we observe that the n-th digit of the result is n − 1, except the ninth one, which is a 9 instead of an 8. But, we can also observe that the n-th partial remainder is n, except the ninth one, which is 0 instead of 9. This can be proved by induction on n: if the n-th partial remainder is n, then the (n + 1)-th partial dividend is 10n + 1, that is 9n + (n + 1). Thus the (n + 1)-th digit of the result is n and (n + 1)-th partial remainder n + 1.
<figure><img src="image_7.png" /><figcaption>Figure 7: The regularity is also in the partial remainders: 1, 2, 3...</figcaption></figure>

12345679 by the product. She then got a raw of nine repetitions of the named digit.

But this explanation is, of course, partial, as the statement

$$12345679 \times 9 = 11111111$$

still remains to be explained.

## 4.2 A deeper explanation

Instead of explaining the statement

$$12345679 \times 9 = 11111111$$

we can attempt to explain the equivalent one

$$11111111/9 = 12345679$$

If we perform the division of 11111111 by 9 (Figure 7), we observe that the n-th digit of the result is n − 1, except the ninth one, which is a 9 instead of an 8. But, we can also observe that the n-th partial remainder is n, except the ninth one, which is 0 instead of 9. This can be proved by induction on n: if the n-th partial remainder is n, then the (n + 1)-th partial dividend is 10n + 1, that is 9n + (n + 1). Thus the (n + 1)-th digit of the result is n and (n + 1)-th partial remainder n + 1.
But to deduce, from the fact that the $(n+1)$-th partial dividend is $9n + (n+1)$, the fact that the $(n+1)$-th digit of the result is $n$ and the $(n+1)$-th partial remainder is $n+1$, we need to use the fact that $n+1 < 9$. This is the case until $n=7$, but for $n=8$ the ninth partial dividend is 81, that does not decompose into $9 \times 8 + 9$, but into $9 \times 9 + 0$. Thus the ninth digit of the result is a 9 and ninth partial remainder 0. It is thus a good point to stop the division.

But this division can also be continued leading to generalize the result: if instead of starting with a raw of nine digits 1, we start with a raw of $9p$ digits 1, we get a result which is a repetition, $p$ times, of the sequence `012345679`. For example, with $p=3$ we get

$$111111111111111111111111/9 = 12345679012345679012345679$$

hence

$$12345679012345679012345679 \times 9 = 11111111111111111111111111111111$$

and in general

$$\left(\sum_{i=0}^{8}(8-i)10^{i} + 1\right) \times \sum_{j=0}^{p-1}10^{9j} \times 9 = \sum_{k=0}^{9p-1}10^{k}$$

This generalization yields another trick: ask the child to multiply the number `12345679012345679012345679` by the product and she will get a more impressive sequence of twenty-seven repetitions of the named digit.

This remark can also be generalized to any base. For example, in base 20, with the digits `0`, `...`, `9`, `a`, `...`, `j`, we have

$$123456789abcdefghj \times j = 1111111111111111111111$$

—note that the $i$ is missing—and more generally, in any base $b^1$

$$\left(\sum_{i=0}^{b-2}(b-2-i)b^i + 1\right) \times \sum_{j=0}^{p-1}b^{(b-1)j} \times (b-1) = \sum_{k=0}^{(b-1)p-1}b^k$$

$$\left(\sum_{i=0}^{b-2}(b-2-i)b^i + 1\right) \times (b-1) = \sum_{i=0}^{b-2}(b-2-i)b^{i+1} - \sum_{i=0}^{b-2}(b-2-i)b^i + b-1 = \sum_{i=0}^{b-2}b^i$$

<footnote n="1">¹If the reader is not convinced by the proof based on the division algorithm, she can check that</footnote>

<footnote>and then multiply both sides by $\sum_{j=0}^{p-1}b^{(b-1)j}$.</footnote>
But to deduce, from the fact that the $(n+1)$-th partial dividend is $9n + (n+1)$, the fact that the $(n+1)$-th digit of the result is $n$ and the $(n+1)$-th partial remainder is $n+1$, we need to use the fact that $n+1 < 9$. This is the case until $n = 7$, but for $n = 8$ the ninth partial dividend is 81, that does not decompose into $9 \times 8 + 9$, but into $9 \times 9 + 0$. Thus the ninth digit of the result is a 9 and ninth partial remainder 0. It is thus a good point to stop the division.

But this division can also be continued leading to generalize the result: if instead of starting with a raw of nine digits 1, we start with a raw of $9p$ digits 1, we get a result which is a repetition, $p$ times, of the sequence `012345679`. For example, with $p = 3$ we get

$$11111111111111111111111111111111/9 = 12345679012345679012345679$$

hence

$$12345679012345679012345679 \times 9 = 11111111111111111111111111111111$$

and in general

$$\left(\sum_{i=0}^{8}(8-i)10^{i} + 1\right) \times \sum_{j=0}^{p-1}10^{9j} \times 9 = \sum_{k=0}^{9p-1}10^{k}$$

This generalization yields another trick: ask the child to multiply the number `12345679012345679012345679` by the product and she will get a more impressive sequence of twenty-seven repetitions of the named digit.

This remark can also be generalized to any base. For example, in base 20, with the digits `0`, `...`, `9`, `a`, `...`, `j`, we have

$$123456789abcdefghj \times j = 111111111111111111111111$$

—note that the $i$ is missing—and more generally, in any base $b^1$

$$\left(\sum_{i=0}^{b-2}(b-2-i)b^i + 1\right) \times \sum_{j=0}^{p-1}b^{(b-1)j} \times (b-1) = \sum_{k=0}^{(b-1)p-1}b^k$$

$$(\sum_{i=0}^{b-2}(b-2-i)b^i + 1) \times (b-1) = \sum_{i=0}^{b-2}(b-2-i)b^{i+1} - \sum_{i=0}^{b-2}(b-2-i)b^i + b-1 = \sum_{i=0}^{b-2}b^i$$

<footnote n="1">¹If the reader is not convinced by the proof based on the division algorithm, she can check that</footnote>

<footnote>and then multiply both sides by $\sum_{j=0}^{p-1} b^{(b-1)j}$.</footnote>
$$ \begin{array}{c} 123456789 \ abcdefghj \\ \hline 3g \\ heb851ifc962jgda74 \\ 369cfj258bei147adh \\ \hline 444444444444444444 \end{array} $$

<caption>Figure 8: At a vigesimal children’s party.</caption>

As the variables $p$ and $b$ can be any natural number, they are implicitly universally quantified in this statement.

This generalization also yields a new magic trick: at a children’s party, in a vigesimal culture, ask a child to name her favorite digit, in your head multiply it by $j$ and ask the child to multiply $123456789abcdefghj$ by the product. She will get a raw of nineteen repetitions of the named digit (Figure 8).

# 5 Explanation: a definition

Instead of proving the statement

$$12345679 \times 36 = 444444444$$

with the multiplication algorithm, we first proved its generalization

$$\forall n \in [1, 9] \ 12345679 \times 9 \times n = 11111111 \times n$$

in a generic way and then deduced the particular case

$$12345679 \times 36 = 444444444$$

As the quantification in the interval $[1, 9]$ is finite, we could have proved each of the nine cases, but this is not how we have proceeded: we have first proved the statement

$$12345679 \times 9 = 11111111$$

and then multiplied both sides by $n$, in a generic way, that is without enumerating the possible cases for $n$.

In the same way, instead of proving the statement

$$12345679 \times 9 = 11111111$$
$$ \begin{array}{c} 123456789 \ abcdefghj \\ \hline 3g \\ heb851ifc962jgda74 \\ 369cfj258bei147adh \\ \hline 444444444444444444 \end{array} $$

<caption>Figure 8: At a vigesimal children’s party.</caption>

As the variables $p$ and $b$ can be any natural number, they are implicitly universally quantified in this statement.

This generalization also yields a new magic trick: at a children’s party, in a vigesimal culture, ask a child to name her favorite digit, in your head multiply it by $j$ and ask the child to multiply $123456789abcdefghj$ by the product. She will get a raw of nineteen repetitions of the named digit (Figure 8).

# 5 Explanation: a definition

Instead of proving the statement

$$12345679 \times 36 = 444444444$$

with the multiplication algorithm, we first proved its generalization

$$\forall n \in [1, 9] \ 12345679 \times 9 \times n = 11111111 \times n$$

in a generic way and then deduced the particular case

$$12345679 \times 36 = 444444444$$

As the quantification in the interval $[1, 9]$ is finite, we could have proved each of the nine cases, but this is not how we have proceeded: we have first proved the statement

$$12345679 \times 9 = 11111111$$

and then multiplied both sides by $n$, in a generic way, that is without enumerating the possible cases for $n$.

In the same way, instead of proving the statement

$$12345679 \times 9 = 11111111$$
using the multiplication or the division algorithm, we have proved the more general statement

$$ (\sum_{i=0}^{8}(8-i)10^{i} + 1) \times \sum_{j=0}^{p-1} 10^{9j} \times 9 = \sum_{k=0}^{9p-1} 10^{k} $$

or even

$$ (\sum_{i=0}^{b-2}(b-2-i)b^{i} + 1) \times \sum_{j=0}^{p-1} b^{(b-1)j} \times (b-1) = \sum_{k=0}^{(b-1)p-1} b^{k} $$

in a generic way—as the implicit universal quantification is on an infinite domain, no enumeration is possible—and then deduced a particular case for $p = 1$ and $b = 10$.

And the more general the intermediate statement, the more explanatory the proof.

This remark leads us to define an explanation of a statement, not merely as a proof of this statement, but as one that proves a general statement in a generic way, and then specialize it to a particular case.

This notion of a proof that proves a general statement in a generic way, and then specializes it to a particular case already exists in proof theory. A general statement is a statement of the form $\forall x A[x]$. A generic proof of such a statement is a proof ending with an introduction rule of the universal quantifier

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \forall\text{-introduction} $$

The specialisation of this proof to a particular case is the proof obtained by adding an elimination rule of the universal quantifier

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \begin{array}{l} \forall\text{-introduction} \\ \forall\text{-elimination} \end{array} \\ \frac{A[t]}{} \end{array} $$

Such a proof ending with an introduction rule of the universal quantifier followed by an elimination rule of this quantifier is called a *cut on the universal quantifier*. It can be contrasted with the proof $\pi[t]$ obtained by substituting
using the multiplication or the division algorithm, we have proved the more general statement

$$ (\sum_{i=0}^{8}(8-i)10^{i} + 1) \times \sum_{j=0}^{p-1} 10^{9j} \times 9 = \sum_{k=0}^{9p-1} 10^{k} $$

or even

$$ (\sum_{i=0}^{b-2}(b-2-i)b^{i} + 1) \times \sum_{j=0}^{p-1} b^{(b-1)j} \times (b-1) = \sum_{k=0}^{(b-1)p-1} b^{k} $$

in a generic way—as the implicit universal quantification is on an infinite domain, no enumeration is possible—and then deduced a particular case for $p = 1$ and $b = 10$.

And the more general the intermediate statement, the more explanatory the proof.

This remark leads us to define an explanation of a statement, not merely as a proof of this statement, but as one that proves a general statement in a generic way, and then specialize it to a particular case.

This notion of a proof that proves a general statement in a generic way, and then specializes it to a particular case already exists in proof theory. A general statement is a statement of the form $\forall x A[x]$. A generic proof of such a statement is a proof ending with an introduction rule of the universal quantifier

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \forall\text{-introduction} $$

The specialisation of this proof to a particular case is the proof obtained by adding an elimination rule of the universal quantifier

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \begin{array}{l} \forall\text{-introduction} \\ \forall\text{-elimination} \end{array} \\ \frac{A[t]}{} \end{array} $$

Such a proof ending with an introduction rule of the universal quantifier followed by an elimination rule of this quantifier is called a *cut on the universal quantifier*. It can be contrasted with the proof $\pi[t]$ obtained by substituting
the generic variable $x$ with the particular term $t$, where the generality of the argument is lost.

Thus, the definition above can be rephrased as the fact that an explanation of a statement is a proof of this statement, that is a cut on the universal quantifier. The proof of the statement

$$12345679 \times 36 = 444444444$$

that proceeds by proving the generalization

$$\forall n \in [1, 9] 12345679 \times 9 \times n = 11111111 \times n$$

in a generic way and then deducing the particular case

$$12345679 \times 36 = 444444444$$

is a cut. That that proceeds by performing the multiplication is not.

This definition of the notion of explanation presupposes that the statement to be explained can be formulated as a particular case of a general statement. The discovery of this general statement is the first step towards an explanation: the first step of the explanation of the statement

$$12345679 \times 36 = 444444444$$

is to remark that 36 and 444444444 are both multiples of 4 hence that this statement can be rephrased

$$12345679 \times 9 \times 4 = 11111111 \times 4$$

and to formulate the hypothesis that, may be, the digit 4 can be replaced by any digit: may be, multiplying 12345679 by the product of any digit by 9 yields a raw of nine repetitions of this digit.

# 6 Revisiting the examples

We can now attempt to confront this definition to the six examples discussed above: the bank loan application, the bisectors of a triangle, the four color theorem, the weather forecast, the data-centric algorithms, and the multiplication of two arbitrary numbers. These examples fall in three categories.
the generic variable $x$ with the particular term $t$, where the generality of the argument is lost.

Thus, the definition above can be rephrased as the fact that an explanation of a statement is a proof of this statement, that is a cut on the universal quantifier. The proof of the statement

$$12345679 \times 36 = 444444444$$

that proceeds by proving the generalization

$$\forall n \in [1, 9] 12345679 \times 9 \times n = 11111111 \times n$$

in a generic way and then deducing the particular case

$$12345679 \times 36 = 444444444$$

is a cut. That that proceeds by performing the multiplication is not.

This definition of the notion of explanation presupposes that the statement to be explained can be formulated as a particular case of a general statement. The discovery of this general statement is the first step towards an explanation: the first step of the explanation of the statement

$$12345679 \times 36 = 444444444$$

is to remark that 36 and 444444444 are both multiples of 4 hence that this statement can be rephrased

$$12345679 \times 9 \times 4 = 11111111 \times 4$$

and to formulate the hypothesis that, may be, the digit 4 can be replaced by any digit: may be, multiplying 12345679 by the product of any digit by 9 yields a raw of nine repetitions of this digit.

# 6 Revisiting the examples

We can now attempt to confront this definition to the six examples discussed above: the bank loan application, the bisectors of a triangle, the four color theorem, the weather forecast, the data-centric algorithms, and the multiplication of two arbitrary numbers. These examples fall in three categories.
In the first category, fall the examples for which we have an explanation, that is a general statement and a generic proof of it. In this category, falls the example of the bisectors of a triangle, for which we first have a generic proof that the bisectors of the sides of any triangles meet at one point, and then a specialization of this proof to the considered triangle.

The example of the bank loan application also falls in this category, when the clerk explains that all applications without a guarantor are rejected. Indeed, in this case we first have a proof that of the general statement “all applications without a guarantor are rejected” and then a specialization of this proof to the considered application. Often the proof of the general statement is a mere reference to a rule of the bank, that can be considered as an axiom.

Note that, if the rule stated that the bank loan approval depends on the gender, the sexual orientation, or the eyes color of the applicant, it would also be an explanation. And such a decision would also be completely transparent. This shows that, from an ethical point of view, the presence of an explanation alone, or transparency alone, is not a sufficient condition for a good action. But the presence of an explanation is a prerequisite to mobilize other values such as gender equality.

In the second category, fall the examples for which we have a general statement, but no a generic proof of it. In this category, falls the example of the four color theorem. This theorem is a consequence of a general statement: all maps belonging to a given set of 1482 maps are reducible. This statement is general, but the only known proof is to enumerate the 1482 maps and check, one by one, that each of them is reducible. This contrasts with the proof of the statement

$$ \forall n \in [1, 9] \quad 12345679 \times 9 \times n = 11111111 \times n $$

that first proves the statement

$$ 12345679 \times 9 = 11111111 $$

and then generically multiplies the two sides by $n$.

Yet a proof with an analysis of 2 cases could still be called explanatory. And a proof with an analysis of 12 cases may be called less explanatory than one with 2 and more explanatory than one with 1482. Finding a threshold between explanatory proofs and non explanatory ones may be as difficult as
In the first category, fall the examples for which we have an explanation, that is a general statement and a generic proof of it. In this category, falls the example of the bisectors of a triangle, for which we first have a generic proof that the bisectors of the sides of any triangles meet at one point, and then a specialization of this proof to the considered triangle.

The example of the bank loan application also falls in this category, when the clerk explains that all applications without a guarantor are rejected. Indeed, in this case we first have a proof that of the general statement “all applications without a guarantor are rejected” and then a specialization of this proof to the considered application. Often the proof of the general statement is a mere reference to a rule of the bank, that can be considered as an axiom.

Note that, if the rule stated that the bank loan approval depends on the gender, the sexual orientation, or the eyes color of the applicant, it would also be an explanation. And such a decision would also be completely transparent. This shows that, from an ethical point of view, the presence of an explanation alone, or transparency alone, is not a sufficient condition for a good action. But the presence of an explanation is a prerequisite to mobilize other values such as gender equality.

In the second category, fall the examples for which we have a general statement, but no a generic proof of it. In this category, falls the example of the four color theorem. This theorem is a consequence of a general statement: all maps belonging to a given set of 1482 maps are reducible. This statement is general, but the only known proof is to enumerate the 1482 maps and check, one by one, that each of them is reducible. This contrasts with the proof of the statement

$$ \forall n \in [1, 9] \quad 12345679 \times 9 \times n = 111111111 \times n $$

that first proves the statement

$$ 12345679 \times 9 = 111111111 $$

and then generically multiplies the two sides by $n$.

Yet a proof with an analysis of 2 cases could still be called explanatory. And a proof with an analysis of 12 cases may be called less explanatory than one with 2 and more explanatory than one with 1482. Finding a threshold between explanatory proofs and non explanatory ones may be as difficult as
finding a threshold between microscopic and macroscopic objects. Yet, this
does not prevent a quark to be microscopic and a whale to be macroscopic.

In the last category, fall the examples for which we do not even have a
general statement. Fall in this category the examples of the weather forecast,
the data-centric algorithms, and the multiplication of two arbitrary numbers.
Because of this lack of a general statement, we do not even see what an
explanation would look like, we do not even see what there is to explain.

In some situations, we try to find such a general statement. For instance,
there is a rule of thumb that, due to dominant west winds, it is often the case
that it rains in Strasbourg one day, if it rained in Paris the day before. If
this statement happened to be true, or even statistically true, then it could
be used as an explanation of the weather in Strasbourg. Unfortunately, the
forecast obtained by using numerical analysis techniques is statistically more
accurate.

In a similar way, if it happened to be true that a dog image more likely
represents a dalmatian than a dachshund, if it contains both a lot of white
pixels and a lot of black pixels. Then, this could be used as an explanation
of the fact that some image represents a dalmatian. In machine learning,
such an attempt to find general statements is one possible direction towards
explanatory artificial intelligence.

Another example: take a database of landscape images, some of them
with a crescent moon and the others with a full moon. Use this database
to train a machine learning algorithm to identify if a new image displays a
crescent moon or a full moon. Then, pointing a small square in the image
displaying a crescent moon is considered as a explanation of the statement
"The picture most likely represents a landscape with a crescent moon". Here
again, pointing to this square in the image is equivalent to formulate a general
statement: "All images containing this square, whatever the rest of the image
is, most likely represent a landscape with a crescent moon".
finding a threshold between microscopic and macroscopic objects. Yet, this
does not prevent a quark to be microscopic and a whale to be macroscopic.

In the last category, fall the examples for which we do not even have a
general statement. Fall in this category the examples of the weather forecast,
the data-centric algorithms, and the multiplication of two arbitrary numbers.
Because of this lack of a general statement, we do not even see what an
explanation would look like, we do not even see what there is to explain.

In some situations, we try to find such a general statement. For instance,
there is a rule of thumb that, due to dominant west winds, it is often the case
that it rains in Strasbourg one day, if it rained in Paris the day before. If
this statement happened to be true, or even statistically true, then it could
be used as an explanation of the weather in Strasbourg. Unfortunately, the
forecast obtained by using numerical analysis techniques is statistically more
accurate.

In a similar way, if it happened to be true that a dog image more likely
represents a dalmatian than a dachshund, if it contains both a lot of white
pixels and a lot of black pixels. Then, this could be used as an explanation
of the fact that some image represents a dalmatian. In machine learning,
such an attempt to find general statements is one possible direction towards
explanatory artificial intelligence.

Another example: take a database of landscape images, some of them
with a crescent moon and the others with a full moon. Use this database
to train a machine learning algorithm to identify if a new image displays a
crescent moon or a full moon. Then, pointing a small square in the image
displaying a crescent moon is considered as a explanation of the statement
"The picture most likely represents a landscape with a crescent moon". Here
again, pointing to this square in the image is equivalent to formulate a general
statement: "All images containing this square, whatever the rest of the image
is, most likely represent a landscape with a crescent moon".
# 7 More examples of explanatory and non explanatory proofs: quantifier elimination

## 7.1 Explanatory proofs that Diophantine equations have no solutions

The Diophantine equation

$$x^2 = 1800$$

has no solutions and the statement expressing it does not

$$\forall x \ x^2 \neq 1800$$

has a simple proof: as $42^2 = 1764$ and $43^2 = 1849$, all numbers smaller than or equal to 42 are too small to be solutions

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$

and all numbers larger than 42 are too large

$$\forall x \ (x > 42 \Rightarrow x^2 > 1800)$$

Hence, using a case analysis, with two cases only, we get a proof of the statement

$$\forall x \ x^2 \neq 1800$$

This proof is explanatory as the proofs of

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$

and of

$$\forall x \ (x > 42 \Rightarrow x^2 > 1800)$$

based on the monotony of the square function, are generic.

## 7.2 Non explanatory proofs that Diophantine equations have no solutions

The statement

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$
# 7 More examples of explanatory and non explanatory proofs: quantifier elimination

## 7.1 Explanatory proofs that Diophantine equations have no solutions

The Diophantine equation

$$x^2 = 1800$$

has no solutions and the statement expressing it does not

$$\forall x \ x^2 \neq 1800$$

has a simple proof: as $42^2 = 1764$ and $43^2 = 1849$, all numbers smaller than or equal to 42 are too small to be solutions

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$

and all numbers larger than 42 are too large

$$\forall x \ (x > 42 \Rightarrow x^2 > 1800)$$

Hence, using a case analysis, with two cases only, we get a proof of the statement

$$\forall x \ x^2 \neq 1800$$

This proof is explanatory as the proofs of

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$

and of

$$\forall x \ (x > 42 \Rightarrow x^2 > 1800)$$

based on the monotony of the square function, are generic.

## 7.2 Non explanatory proofs that Diophantine equations have no solutions

The statement

$$\forall x \ (x \leq 42 \Rightarrow x^2 < 1800)$$
unlike

$$ \forall x (x > 42 \Rightarrow x^2 > 1800) $$

uses a quantification over a finite domain. Hence it can also be proved by a simple enumerations of the 43 cases $x = 0, x = 1, ..., x = 42$.

Because of this enumeration, the obtained proof is not explanatory anymore. But this method can be generalized to any univariate Diophantine equation

$$ a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0 $$

When $x$ is larger than $b = n \max(|a_0|, ..., |a_{n-1}|)$, then $|a_n x^n|$ is larger than $|a_{n-1} x^{n-1} + \dots + a_1 x + a_0|$ and $x$ thus cannot be a solution. Therefore, the statement

$$ \exists x (a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0) $$

is equivalent to

$$ \exists x (x \le b \land a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0) $$

where the infinitude of all possible natural numbers has been reduced to $b+1$, that can be checked one by one.

This method called quantifier elimination because the existential quantifier is replaced with a bounded one and then with an enumeration, always produces non explanatory proofs.

Besides the non existence of solutions of univariate Diophantine equations, quantifier elimination is used in the proof of the four color theorem, and in the proofs built with Presburger's, Skolem's, and Tarski's algorithms.

# 8 From proofs to algorithms

In contemporary logic, the notion of proof is not a primitive notion anymore, but after Brouwer, Heyting, and Kolmogorov, it has been defined in terms of a more primitive one: that of algorithm.

In particular a proof of a statement of the form $\forall x A[x]$ is an algorithm mapping every term $u$ to a proof of $A[u]$. The proof

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \forall\text{-introduction} $$
unlike

$$ \forall x (x > 42 \Rightarrow x^2 > 1800) $$

uses a quantification over a finite domain. Hence it can also be proved by a simple enumerations of the 43 cases $x = 0, x = 1, ..., x = 42$.

Because of this enumeration, the obtained proof is not explanatory anymore. But this method can be generalized to any univariate Diophantine equation

$$ a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0 $$

When $x$ is larger than $b = n \max(|a_0|, ..., |a_{n-1}|)$, then $|a_n x^n|$ is larger than $|a_{n-1} x^{n-1} + \dots + a_1 x + a_0|$ and $x$ thus cannot be a solution. Therefore, the statement

$$ \exists x (a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0) $$

is equivalent to

$$ \exists x (x \le b \land a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0) $$

where the infinitude of all possible natural numbers has been reduced to $b+1$, that can be checked one by one.

This method called quantifier elimination because the existential quantifier is replaced with a bounded one and then with an enumeration, always produces non explanatory proofs.

Besides the non existence of solutions of univariate Diophantine equations, quantifier elimination is used in the proof of the four color theorem, and in the proofs built with Presburger's, Skolem's, and Tarski's algorithms.

# 8 From proofs to algorithms

In contemporary logic, the notion of proof is not a primitive notion anymore, but after Brouwer, Heyting, and Kolmogorov, it has been defined in terms of a more primitive one: that of algorithm.

In particular a proof of a statement of the form $\forall x A[x]$ is an algorithm mapping every term $u$ to a proof of $A[u]$. The proof

$$ \frac{\frac{\pi[x]}{A[x]}}{\forall x A[x]} \quad \forall\text{-introduction} $$
is the algorithm mapping the term $u$ to the proof $\pi[u]$.

The explanation of $A[t]$

$$ \frac{\frac{\pi[x]}{A[x]}}{\frac{\forall x A[x]}{A[t]}} \quad \begin{array}{l} \forall\text{-introduction} \\ \forall\text{-elimination} \end{array} $$

thus contains an algorithm: that mapping the term $u$ to the proof $\pi[u]$, and an input value for this algorithm: the term $t$.

This leads to define, more generally, an explanation as a pair formed with an algorithm and an input value for this algorithm. When this algorithm is applied to this value, it returns a proof of the explained statement.

According to Brouwer, Heyting, and Kolmogorov, a proof of a statement of the form

$$ \forall x (A[x] \Rightarrow \exists y B[y]) $$

is an algorithm mapping each pair formed with a term $t$ and a proof of the statement $A[t]$ to a pair formed with a term $u$ and a proof of the statement $B[u]$. If the proofs of the statements of the form $A[\cdot]$ and $B[\cdot]$ are trivial, for example if these statements belong to a decidable fragment or if they can be directly observed, then a proof of this statement is simply an algorithm mapping each term $t$ such that $A[t]$ to a term $u$ such that $B[u]$, this term $u$ playing the rôle of a proof of $\exists x B[x]$. Then, an explanation of the statement $\exists y B[y]$ is simply a pair formed with

* an algorithm mapping each term $t$ such that $A[t]$ to a term $u$ such that $B[u]$

* and an input value for this algorithm, that is a term $t$ such that $A[t]$.

This notion gets closer to the usual notion of explanation. If $A[x]$ is the statement “the order $x$ has been placed yesterday” and $B[y]$ is the statement “the book $y$ has been delivered by the postman today”. Then, an explanation of the statement $\exists y B[y]$：“a book has been delivered by the postman today” is a pair formed with an algorithm mapping orders placed to books delivered— such an algorithm may be called a *book shop*—and an order that has been placed: a book has been delivered today because an order has been placed yesterday at the book shop.
is the algorithm mapping the term $u$ to the proof $\pi[u]$.

The explanation of $A[t]$

$$ \frac{\frac{\pi[x]}{A[x]}}{\frac{\forall x A[x]}{A[t]}} \quad \begin{array}{l} \forall\text{-introduction} \\ \forall\text{-elimination} \end{array} $$

thus contains an algorithm: that mapping the term $u$ to the proof $\pi[u]$, and an input value for this algorithm: the term $t$.

This leads to define, more generally, an explanation as a pair formed with an algorithm and an input value for this algorithm. When this algorithm is applied to this value, it returns a proof of the explained statement.

According to Brouwer, Heyting, and Kolmogorov, a proof of a statement of the form

$$ \forall x (A[x] \Rightarrow \exists y B[y]) $$

is an algorithm mapping each pair formed with a term $t$ and a proof of the statement $A[t]$ to a pair formed with a term $u$ and a proof of the statement $B[u]$. If the proofs of the statements of the form $A[\cdot]$ and $B[\cdot]$ are trivial, for example if these statements belong to a decidable fragment or if they can be directly observed, then a proof of this statement is simply an algorithm mapping each term $t$ such that $A[t]$ to a term $u$ such that $B[u]$, this term $u$ playing the rôle of a proof of $\exists x B[x]$. Then, an explanation of the statement $\exists y B[y]$ is simply a pair formed with

* an algorithm mapping each term $t$ such that $A[t]$ to a term $u$ such that $B[u]$

* and an input value for this algorithm, that is a term $t$ such that $A[t]$.

This notion gets closer to the usual notion of explanation. If $A[x]$ is the statement “the order $x$ has been placed yesterday” and $B[y]$ is the statement “the book $y$ has been delivered by the postman today”. Then, an explanation of the statement $\exists y B[y]$：“a book has been delivered by the postman today” is a pair formed with an algorithm mapping orders placed to books delivered— such an algorithm may be called a *book shop*—and an order that has been placed: a book has been delivered today because an order has been placed yesterday at the book shop.
Note that the book itself is a proof of the statement “a book has been delivered”, but it is not an explanation of this statement. In contrast, the pair formed with the book shop and the order placed is such an explanation.

# 9 The weather forecast paradox

We may wonder if this definition of an explanation as a pair formed with an algorithm and an input value for this algorithm is not too general. In particular, as the weather forecast of Figure 3 is obtained by applying an algorithm performing numerical analysis computation to sensor data, the pair formed with the algorithm and the sensor data would, according to our definition, be an explanation of the forecast, even if we do not see a general statement, this forecast would be a particular case of.

But, as already remarked, this proof is not explanatory, not because it is not formed with an algorithm and an input value for this algorithm, but because the program expressing this algorithm is large and the execution of this algorithm is long. So, nobody can trace the computation step by step. This situation can be contrasted with that of the explanation of the statement

$$12345679 \times 9 \times 4 = 11111111 \times 4$$

or even the statement

$$12345679 \times 36 = 44444444$$

that is formed with a proof of the statement

$$\forall n \in [1, 9] 12345679 \times 9 \times n = 11111111 \times n$$

and the input value 4. In this case, the algorithm is expressed by a short program and its execution is fast.

This situation is common in ethics: if a clerk refuses a bank loan to a customer, not only she ought to explain her decision, but she ought to explain her decision in a way that can be understood by the customer, for example, in a language that the customer speaks. This is why, for example, if a person is sued in a country she does not understand the language of, the tribunal ought to provide an interpreter.

This means that the quality of the explanation does not only depend on the width of the possible input values of the algorithm: the generality of the
Note that the book itself is a proof of the statement “a book has been delivered”, but it is not an explanation of this statement. In contrast, the pair formed with the book shop and the order placed is such an explanation.

# 9 The weather forecast paradox

We may wonder if this definition of an explanation as a pair formed with an algorithm and an input value for this algorithm is not too general. In particular, as the weather forecast of Figure 3 is obtained by applying an algorithm performing numerical analysis computation to sensor data, the pair formed with the algorithm and the sensor data would, according to our definition, be an explanation of the forecast, even if we do not see a general statement, this forecast would be a particular case of.

But, as already remarked, this proof is not explanatory, not because it is not formed with an algorithm and an input value for this algorithm, but because the program expressing this algorithm is large and the execution of this algorithm is long. So, nobody can trace the computation step by step. This situation can be contrasted with that of the explanation of the statement

$$12345679 \times 9 \times 4 = 11111111 \times 4$$

or even the statement

$$12345679 \times 36 = 44444444$$

that is formed with a proof of the statement

$$\forall n \in [1, 9] 12345679 \times 9 \times n = 11111111 \times n$$

and the input value 4. In this case, the algorithm is expressed by a short program and its execution is fast.

This situation is common in ethics: if a clerk refuses a bank loan to a customer, not only she ought to explain her decision, but she ought to explain her decision in a way that can be understood by the customer, for example, in a language that the customer speaks. This is why, for example, if a person is sued in a country she does not understand the language of, the tribunal ought to provide an interpreter.

This means that the quality of the explanation does not only depend on the width of the possible input values of the algorithm: the generality of the
statement, but also on the small size of the program expressing the algorithm and the low complexity, that is the small execution time, of this algorithm. As these notions—small, fast—are quantitative rather than qualitative, the notion of explanation also is.

So it is possible to order explanations: the shorter the program and the faster its execution, the deeper the explanation. Yet, it is also possible to identify thresholds. For instance, using the notion of Kolmogorov complexity, we can call the pair formed with an algorithm and an input value an explanation if the size of the program expressing the algorithm and of the data expressing the input value is smaller than the size of the explained statement.

# 10 From proof to explanation

Checking the 1482 cases of the four color theorem, applying the finite element method, using a data-centric algorithm, checking the statement

$$ \forall x (x \le 42 \Rightarrow x^2 \ne 1800) $$

or using Tarski's algorithm with a pencil and a paper, and even with a pocket calculator, is tedious, and sometimes impossible. The size of the data, the size of the program expressing the algorithm, the complexity of the algorithm... require the use of a computer.

The same size of data, size of the program, and complexity of this algorithm make the proof non explanatory. This explains that non explanatory proofs have been built mostly after the invention of the computer.

The use of computers both made possible the construction of many non explanatory proofs, and the need of a formal definition of the notion of explanation.

We propose here a first attempt towards such a definition: a pair formed with a small and fast algorithm and an input value to this algorithm such that the algorithm applied to the value produces a proof of the statement to be explained. So, the articulation between proof and explanation and that between computability and complexity are parallel.

Philosophy of computer science first focused on epistemological questions, then ethical questions were also raised. The notion of proof—whether explanatory or not—was the right tool to address many epistemological questions, but it seems that the notion of explanation is needed to address the ethical ones.
statement, but also on the small size of the program expressing the algorithm and the low complexity, that is the small execution time, of this algorithm. As these notions—small, fast—are quantitative rather than qualitative, the notion of explanation also is.

So it is possible to order explanations: the shorter the program and the faster its execution, the deeper the explanation. Yet, it is also possible to identify thresholds. For instance, using the notion of Kolmogorov complexity, we can call the pair formed with an algorithm and an input value an explanation if the size of the program expressing the algorithm and of the data expressing the input value is smaller than the size of the explained statement.

# 10 From proof to explanation

Checking the 1482 cases of the four color theorem, applying the finite element method, using a data-centric algorithm, checking the statement

$$ \forall x (x \le 42 \Rightarrow x^2 \ne 1800) $$

or using Tarski's algorithm with a pencil and a paper, and even with a pocket calculator, is tedious, and sometimes impossible. The size of the data, the size of the program expressing the algorithm, the complexity of the algorithm... require the use of a computer.

The same size of data, size of the program, and complexity of this algorithm make the proof non explanatory. This explains that non explanatory proofs have been built mostly after the invention of the computer.

The use of computers both made possible the construction of many non explanatory proofs, and the need of a formal definition of the notion of explanation.

We propose here a first attempt towards such a definition: a pair formed with a small and fast algorithm and an input value to this algorithm such that the algorithm applied to the value produces a proof of the statement to be explained. So, the articulation between proof and explanation and that between computability and complexity are parallel.

Philosophy of computer science first focused on epistemological questions, then ethical questions were also raised. The notion of proof—whether explanatory or not—was the right tool to address many epistemological questions, but it seems that the notion of explanation is needed to address the ethical ones.
Thus, the articulations between epistemology and ethics, between proof and explanation, and between computability and complexity are parallel.

## References

[1] K. Appel and W. Haken. Every planar map is four colorable. i. discharging. *Illinois Journal of Mathematics*, 21(3):429–490, 1977.

[2] K. Appel, W. Haken, and J. Koch. Every planar map is four colorable. ii. reducibility. *Illinois Journal of Mathematics*, 21(3):491–567, 1977.

[3] G. Dowek. *Computation, Proof, Machine: Mathematics Enters a New Age*. Cambridge University Press, 2015.

[4] M. Gardner. *The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays*. Springer, 1996.
Thus, the articulations between epistemology and ethics, between proof and explanation, and between computability and complexity are parallel.

## References

[1] K. Appel and W. Haken. Every planar map is four colorable. i. discharging. *Illinois Journal of Mathematics*, 21(3):429–490, 1977.

[2] K. Appel, W. Haken, and J. Koch. Every planar map is four colorable. ii. reducibility. *Illinois Journal of Mathematics*, 21(3):491–567, 1977.

[3] G. Dowek. *Computation, Proof, Machine: Mathematics Enters a New Age*. Cambridge University Press, 2015.

[4] M. Gardner. *The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays*. Springer, 1996.