This thesis highlights two sides of an important and not previously discussed issue-the relationship between Zubiri's philosophy and mathematics:
1. Zubiri's philosophy originates in large measure from the need to lay the foundations of mathematics and to interpret its results. This logico-mathematical involvement in the genesis and development of Zubiri's philosophy is essential.
2. Zubiri's thought implies a mathematical philosophy, a constructivism, which gives an original answer to the problem of the nature and ground of mathematics.
The present thesis comprises five chapters:
I. Mathematics: a suggestion in regard to Zubirian philosophy,
II. Zubiri's Noology: a condition for mathematical constructivism,
III. Sensing constructivism: mathematical intellection,
IV. Transcendental constructivism: the mathematical object,
V. Logico-histoncal constructivism: the mathematical truth.
It contains following results.
Gödel's Theorem (1931) has deep repercusions in Zubiri (as in Lakatos, Quine and a number of other philosophers of mathematics). It represented a great shock to both his ideal-objectivistic philosophy of mathematics and his objectivistic philosophy, and it led him to a realistic interpretation of mathematics and to its underlying philosophy of Reality. This is the cornerstone of his Trilogy about intellection, Sentient Intelligence: "noology" and mathematical philosophy.
The symbiosis between Zubiri's philosophy and Gödelian mathematics gives birth to a new mathematical realism.
The notion of a mathematical function opens a philosophical horizon for Zubiri (as for Russell). It suggests to him a paradigmatic change in the notions of cause, substantial subject, and predicative description, which are to be replaced by those of function, functional structure and functional description.
His mathematical constructivism, which originates from the sentient intelligence and the impression of formality of reality, presupposes a "noology". Its achievement with respect to other philosophies of mathematics, which originate from content and the conceiving intelligence is the nonsynthetic, intrinsic unity that it establishes between apparently irreconcilable terms. The mathematical construction is at one and the same time, pro indiviso, to create and to sense, freedom and imposition, construction and reality, deduction and experience, construction and truth, fulfillment and encounter, and logical and historical. This connection is a milestone in the history of mathematical philosophy.
Zubiri marks the boundary of a "kingdom" of the mathematical things and of all free things, a transcendental ambient of reality in impression, different from the platonic world, both physical and mental. Its respectivity grounds the regressus ad infinitum or mathematical inexhaustibility (Gödel's Theorem).
A comparison of Zubiri's constructivism with other mathematical philosophies (Kantian constructivism, Russell's logicism, Hilbert's formalism. Brouwer's intuitionism, Gödel's realism and Lakato's fallibilism) discloses:
1. The radicalness of its perspective: it grounds and organizes the rest.
2. It overcomes the difficulties of Kantian constructivism: it is an infinitism, a logical historicism and it explain non-euclidean geometries as real constructions.
3. Its congruence with Gödel's new philosophy of mathematics: mathematics is the science of reality and experience, similar to physical sciences and unfeasible for a machine.
4. It provides a non-dogmatic foundation for mathematics, which avoids scepticism and justifies its "advance".