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Husserl’s Conceptions of Formal Mathematics

Edmund Husserl’s conception of mathematics was a unique blend of Platonist and formalist ideas. He believed that mathematics had reached a mixed state combining Platonic and formal elements and that both were important for the pursuit of the sciences, as well as for each other. However, he seemed to believe that only the Platonic aspects had significance for his science of phenomenology. Because of the significance of the distinction between these two types of mathematics, I will always use one of the adjectives “material” or “formal” when discussing any branch of mathematics, unless I specifically mean to include both.

First, I must specify what I mean by each of these terms. By material mathematics, I will mean mathematics as it had traditionally been done before the conceptions of imaginary numbers and non-Euclidean geometry. Thus, any branch of material mathematics seeks to describe how some class of existing things actually behaves. So material geometry seeks to describe how objects lie in space, material number theory seeks to describe how the actual natural numbers are related, and material logic seeks to describe how concepts actually relate to one another. Some of these areas (like material geometry) seek to deal with the physical world, while others (like material logic) deal with abstract objects, so I avoid using the word “Platonic”, which suggests only the latter. By formal mathematics, I will mean mathematics done as is typical in the 20th century, purely axiomatically, without regard to what sorts of objects it might actually describe. Thus, for formal geometry it is irrelevant whether the objects described are physical objects in actual space, or n-tuples of real nu...

... middle of paper ...

... Bouvier, Bonn, 1981.

Tieszen, Richard L. “Mathematical Intuition: Phenomenology and Mathematical Knowledge”. Kluwer, Boston, 1989.

Zalta, Ed. “Frege’s Logic, Theorem and Foundations for Arithmetic”. Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/frege-logic/

Footnotes

1. Lohmar, p. 14

2. However, this claim is itself a material claim of the truth of a statement in material logic, i.e. that the given statement follows from the given axioms, when this statement and these axioms are viewed as actual objects in our reasoning system.

3. Husserl, p. 16

4. Føllesdal, in Hintikka, p. 442

5. Hill, p. 153

6. Husserl, p. xxiii

7. Husserl, p. 161

8. Gödel, p. 385

9. Husserl, p. 163-4

10. Husserl, p. 167-8

11. Husserl, p. 169

12. Husserl, p. 168-9

13. Husserl, p. 136

14. Gödel, p. 385

15. See Zalta’s discussion of Basic Law V.

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Edmund Husserl’s conception of mathematics was a unique blend of Platonist and formalist ideas. He believed that mathematics had reached a mixed state combining Platonic and formal elements and that both were important for the pursuit of the sciences, as well as for each other. However, he seemed to believe that only the Platonic aspects had significance for his science of phenomenology. Because of the significance of the distinction between these two types of mathematics, I will always use one of the adjectives “material” or “formal” when discussing any branch of mathematics, unless I specifically mean to include both.

First, I must specify what I mean by each of these terms. By material mathematics, I will mean mathematics as it had traditionally been done before the conceptions of imaginary numbers and non-Euclidean geometry. Thus, any branch of material mathematics seeks to describe how some class of existing things actually behaves. So material geometry seeks to describe how objects lie in space, material number theory seeks to describe how the actual natural numbers are related, and material logic seeks to describe how concepts actually relate to one another. Some of these areas (like material geometry) seek to deal with the physical world, while others (like material logic) deal with abstract objects, so I avoid using the word “Platonic”, which suggests only the latter. By formal mathematics, I will mean mathematics done as is typical in the 20th century, purely axiomatically, without regard to what sorts of objects it might actually describe. Thus, for formal geometry it is irrelevant whether the objects described are physical objects in actual space, or n-tuples of real nu...

... middle of paper ...

... Bouvier, Bonn, 1981.

Tieszen, Richard L. “Mathematical Intuition: Phenomenology and Mathematical Knowledge”. Kluwer, Boston, 1989.

Zalta, Ed. “Frege’s Logic, Theorem and Foundations for Arithmetic”. Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/frege-logic/

Footnotes

1. Lohmar, p. 14

2. However, this claim is itself a material claim of the truth of a statement in material logic, i.e. that the given statement follows from the given axioms, when this statement and these axioms are viewed as actual objects in our reasoning system.

3. Husserl, p. 16

4. Føllesdal, in Hintikka, p. 442

5. Hill, p. 153

6. Husserl, p. xxiii

7. Husserl, p. 161

8. Gödel, p. 385

9. Husserl, p. 163-4

10. Husserl, p. 167-8

11. Husserl, p. 169

12. Husserl, p. 168-9

13. Husserl, p. 136

14. Gödel, p. 385

15. See Zalta’s discussion of Basic Law V.

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