Est-ce que c'est pareil de faire 2+2 avec des nombres ou avec des pommes ?
Philosophy of arithmetic

Popper's principle of falsifiability runs into prima facie difficulties when the epistemological status of mathematics is considered. It is difficult to conceive how simple statements of arithmetic, such as "2 + 2 = 4", could ever be shown to be false. If they are not open to falsification they can not be scientific. If they are not scientific, it needs to be explained how they can be informative about real world objects and events.

Popper's solution was an original contribution in the philosophy of mathematics. His idea was that a number statement such as "2 apples + 2 apples = 4 apples" can be taken in two senses. In its pure mathematics sense, "2 + 2 = 4" is logically true and cannot be refuted. Contrastingly, in its applied mathematics sense of it describing the physical behaviour of apples, it can be falsified. This can be done by placing two apples in a container, then proceeding to place another two apples in the same container. If there are five, three, or a number of apples that is not four in said container, the theory that "2 apples + 2 apples = 4 apples" is shown to be false. On the contrary, if there are four apples in the container, the theory of numbers is shown to be applicable to reality.
Multiplication de différents types de nombres

Les nombres peuvent compter (3 pommes), ordonner (la 3e pomme) ou mesurer (3,5 pieds de haut) ; au fur et à mesure que l'histoire des mathématiques a progressé du comptage sur les doigts à la modélisation de la mécanique quantique, la multiplication a été généralisée à des types de nombres plus complexes et abstraits, et à des choses qui ne sont pas des nombres (comme les matrices) ou qui ne ressemblent pas beaucoup à des nombres (comme les quaternions).
### Query ###
Est-ce que c'est pareil de faire 2+2 avec des nombres ou avec des pommes ?

### Main Text ###
Philosophy of arithmetic

Popper's principle of falsifiability runs into prima facie difficulties when the epistemological status of mathematics is considered. It is difficult to conceive how simple statements of arithmetic, such as "2 + 2 = 4", could ever be shown to be false. If they are not open to falsification they can not be scientific. If they are not scientific, it needs to be explained how they can be informative about real world objects and events.

Popper's solution was an original contribution in the philosophy of mathematics. His idea was that a number statement such as "2 apples + 2 apples = 4 apples" can be taken in two senses. In its pure mathematics sense, "2 + 2 = 4" is logically true and cannot be refuted. Contrastingly, in its applied mathematics sense of it describing the physical behaviour of apples, it can be falsified. This can be done by placing two apples in a container, then proceeding to place another two apples in the same container. If there are five, three, or a number of apples that is not four in said container, the theory that "2 apples + 2 apples = 4 apples" is shown to be false. On the contrary, if there are four apples in the container, the theory of numbers is shown to be applicable to reality.

### Supplementary Text ###
Multiplication de différents types de nombres

Les nombres peuvent compter (3 pommes), ordonner (la 3e pomme) ou mesurer (3,5 pieds de haut) ; au fur et à mesure que l'histoire des mathématiques a progressé du comptage sur les doigts à la modélisation de la mécanique quantique, la multiplication a été généralisée à des types de nombres plus complexes et abstraits, et à des choses qui ne sont pas des nombres (comme les matrices) ou qui ne ressemblent pas beaucoup à des nombres (comme les quaternions).

### Draft ###

### 1. Query Parsing

"Est-ce que c'est pareil de faire 2+2 avec des nombres ou avec des pommes ?"

Core question: Are mathematical operations identical regardless of object type? 

Translation nuance: "2+2" could mean "2+2=4" (operation) or "two things plus two things" (semantic). User likely asking about operational equivalence.

### 2. Initial Assessment

Simple information retrieval? No. Requires philosophical synthesis.

Domains needed:
- Philosophy of mathematics ●
- Mathematical realism ◐  
- Cognitive science (number representation) ◐
- Applied vs pure mathematics ●

### 3. Core Mathematical Context

Classical view: Mathematics is purely logical. 2+2=4 is analytic truth, true by definition.

But! Applied mathematics operates differently. "2 apples + 2 apples" is empirical claim about physical objects.

Key distinction:
- Pure: 2+2=4 (logical/formal)
- Applied: 2 objects + 2 objects = 4 objects (empirical)

### 4. Philosophical Framework

Immanuel Kant: Mathematics operates via pure intuition (space) and pure understanding (time). Numbers exist a priori in mind.

Logical positivists: Mathematical truths are analytic, follow from definitions.

Quine-Duhem thesis: Mathematics empirically entangled with physical theory.

### 5. Cognitive Science Angle

Developmental psychology: Young children count objects before abstract numbers.

Neuroscience: Different brain regions process numerical magnitudes vs object counting.

∴ Intuitive mathematics emerges from embodied experience.

### 6. Applied Mathematics Reality

"2+2" as pure logic: Always true in formal systems.

"2 apples + 2 apples" as applied: 
- Physical constraints matter
- Measurement error possible  
- Counting failures occur
- Context-dependent interpretation

### 7. Multiplicative Extensions

⟨H≈0.8⟩ Mathematics expands outward from fingers and stones... three apples times two baskets becomes matrix multiplication in quantum field theory... the same structural relationships, different substrates... 3×2=6 whether you're counting fruit or calculating in Hilbert space... patterns transcend embodiment...

Back to focus: ●

Multiplication examples:
- 3 pommes × 2 paniers = 6 fruits
- 3 objects × 2 transformations = 6 possible states  
- 3 matrices × 2 operations = 6-dimensional result space

### 8. Synthesis

Answer structure:
1. Operational equivalence: Yes, for pure math
2. Empirical equivalence: No, for applied math  
3. Cognitive foundation: Embodied→abstract
4. Mathematical expansion: Same patterns, different domains

Key insight: Mathematics contains both logical truths and empirical applications. The "sameness" depends on interpretation level.

### 9. Response Construction

Lead with operational similarity (addition/multiplication commute, associate). 

Distinguish pure vs applied mathematics. Pure: logically necessary. Applied: empirically contingent.

Connect to mathematical philosophy (Kant, Quine).

Extend to complex numbers, matrices, quaternions. Same operations, different mathematical structures.

∴ Qualified yes: Pure mathematics identical across contexts. Applied mathematics context-dependent but follows same logical framework.

### Answer###
En mathématiques, faire 2+2 avec des nombres purs ou avec des objets concrets comme des pommes présente des différences subtiles mais importantes.

D'un point de vue mathématique pur, l'opération "2+2=4" est considérée comme une vérité logique qui ne peut pas être contredite. Cette équation fonctionne dans le domaine des mathématiques pures où les nombres sont des concepts abstraits.

Cependant, lorsque vous appliquez cette même opération aux pommes réelles, cela devient une question différente. L'énoncé "2 pommes + 2 pommes = 4 pommes" appartient au domaine des mathématiques appliquées et peut effectivement être vérifié ou falsifié par l'expérience. Vous pouvez le tester en plaçant deux pommes dans un contenant, puis en ajoutant deux autres pommes. Si vous obtenez cinq, trois ou un nombre différent de quatre pommes, la théorie est fausse.

Cette distinction reflète la philosophie des mathématiques selon laquelle les mathématiques pures consistent en énoncés sans faille qui ne peuvent être contredits, tandis que leurs applications pratiques peuvent être confrontées à la réalité.

Concernant la multiplication, le principe s'étend : vous pouvez multiplier 3 pommes par 2 paniers pour obtenir 6 fruits, et ce même concept s'applique aux objets non numériques comme les matrices ou aux structures mathématiques complexes comme les quaternions.