## The Foundational Crisis of Mathematics and Its Evolution
### Introduction
In the same period, it appeared in various areas of mathematics that the former intuitive definitions of basic mathematical objects were insufficient for ensuring mathematical rigor. Examples of such intuitive definitions include "a set is a collection of objects," "a natural number is what is used for counting," "a point is a shape with zero length in every direction," and "a curve is a trace left by a moving point." This realization marked the origin of the foundational crisis of mathematics. This crisis was eventually addressed by systematizing the axiomatic method within a formalized set theory.
### The Axiomatic Method and Formalized Set Theory
The axiomatic method involves defining mathematical objects by a set of similar objects and the properties they must have. For instance, in Peano arithmetic, natural numbers are defined by axioms such as "zero is a number," "each number has a unique successor," and "each number but zero has a unique predecessor," along with specific rules of reasoning (Peano, 1889). The philosophical nature of these objects is a topic left to philosophers, although many mathematicians have their own opinions or "intuition" to guide their studies and proofs.
### Gödel’s Incompleteness Theorems
This approach allows considering "logics," theorems, proofs, etc., as mathematical objects and enables the proving of theorems about them. Gödel's incompleteness theorems, for example, assert that in every theory containing natural numbers, there exist theorems that are true but not provable within the theory (Gödel, 1931). This profound result highlighted the limitations of formal systems and had a significant impact on the philosophy of mathematics.
### Intuitionistic Logic and Mathematical Logic
The foundational approach of mathematics faced challenges in the early 20th century, notably from L. E. J. Brouwer and his promotion of intuitionistic logic, which excludes the law of excluded middle (Brouwer, 1908). These debates spurred a wide expansion in mathematical logic, leading to the development of subareas such as model theory, proof theory, type theory, computability theory, and computational complexity theory. Although these aspects were introduced before the advent of computers, they significantly influenced computer science, particularly in compiler design, program certification, and proof assistants.
### Applied Mathematics
Applied mathematics focuses on mathematical methods used in science, engineering, business, and industry. It is a mathematical science with specialized knowledge, where professionals work on practical problems and use mathematical models in various fields. Historically, practical applications have often driven the development of mathematical theories, which later become subjects of pure mathematics. Hence, applied mathematics is deeply connected with research in pure mathematics (Courant & Hilbert, 1937).
### Statistics and Decision Sciences
Statistics, closely related to applied mathematics, formulates its theory mathematically, particularly with probability theory. Statisticians design experiments and analyze data to make sense of observations, using modeling and inference theory. Statistical decision problems involve minimizing objective functions under specific constraints, sharing concerns with other decision sciences such as operations research, control theory, and mathematical economics (Fisher, 1925).
### Computational Mathematics
Computational mathematics studies methods for solving mathematical problems that exceed human numerical capacity. Numerical analysis, a significant area within this field, investigates approximation and discretization methods, focusing on rounding errors. Scientific computing extends these methods to non-analytic topics, such as algorithmic matrix and graph theory, and includes computer algebra and symbolic computation (Trefethen & Bau, 1997).
### Historical Development of Mathematics
The history of mathematics is marked by increasing abstraction. Early mathematics involved recognizing numerical quantities, as evidenced by prehistoric tallies. By 3000 BC, Babylonians and Egyptians employed arithmetic, algebra, and geometry for various practical purposes. Greek mathematics began around the 6th century BC, with systematic studies and the axiomatic method introduced by Euclid around 300 BC. Archimedes of Syracuse made significant contributions with formulas for areas and volumes of solids and the method of exhaustion, precursors to calculus (Heath, 1921).
During the Golden Age of Islam, significant advancements in algebra, trigonometry, and the decimal system were made. The early modern period saw accelerated development in Western Europe, with calculus' invention by Newton and Leibniz. The 19th century brought rigorous studies and abstract topics, and the 20th century saw Gödel's incompleteness theorems, reshaping the landscape of mathematical logic (Rashed, 1994).
### Conclusion
Mathematics continues to evolve, with ongoing discoveries and fruitful interactions between mathematics and other sciences. The aesthetic and practical aspects of mathematics underscore its significance and enduring appeal.
### References
Brouwer, L. E. J. (1908). De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte.
Courant, R., & Hilbert, D. (1937). Methoden der mathematischen Physik. Springer.
Fisher, R. A. (1925). Statistical Methods for Research Worker. Oliver and Boyd.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173-198.
Heath, T. L. (1921). A History of Greek Mathematics. Clarendon Press.
Peano, G. (1889). Arithmetices principia, nova methodo exposita. Fratres Bocca.
Rashed, R. (1994). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer.
Trefethen, L. N., & Bau, D. (1997). Numerical Linear Algebra. SIAM.
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