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Mathematics
University
Introduction to the Bachelor's Degree in Mathematics. Source: Wikipedia (CC BY-SA 4.0).
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What you need to know
Read the full theory →Definition, description, and history of mathematics
- Definition: Formal science of patterns, structures, and relations in logical systems.
- Objects of study: Quantity, shape, space, and number.
- Basis of reasoning: Abstraction and logic.
- Applied mathematics: Use of knowledge in other fields and the creation of new disciplines.
- Pure mathematics: Theoretical study without immediate applied purposes.
- Hellenic Greece: Origin of logical proof and Euclidean geometry.
- René Descartes: Introduction of coordinates and computational approach.
- Leibniz and Isaac Newton: Development of infinitesimal calculus.
- Niels Henrik Abel and Évariste Galois: Development of the concept of groups (symmetries).
- Andrei Kolmogorov: Publication of the foundations of probability theory in 1933.
- Cauchy and Weierstrass: Rigorization of infinitesimal calculus in the 19th century.
- Georg Cantor: Development of set theory.
- 20th Century: Focus on axiomatization and the abstraction of essential properties.
- Category theory: A common framework for mathematical constructions across different areas.
Etymology, definitions, and epistemology of mathematics
- Etymology: mathēmatiká means "things that are learned"
- Distinction: Mathematical art (requires instruction) vs. Art of the muses (direct appreciation)
- Technical consolidation: Carried out by Aristotle in the 4th century BC.
- Bourbaki debate: Conceptual uniqueness of mathematics versus its plural form
- Mathematical objective: Search for patterns and conjectures through rigorous deductions
- Nature: Tautological, infallible, and a priori (unlike the natural sciences)
- Function: Formal language for the study of structures, magnitudes, and quantitative relations
- Application: Real, complex, and matrix analysis arise from the demands of physics and chemistry
History, nature, aesthetics, and rigor of mathematics
- Origin: Calculus predates writing, linked to accounting and surveying.
- Scientific interaction: Mathematics and natural sciences generate mutual problems (e.g., string theory).
- Efficacy: Concept of the "unreasonable effectiveness of mathematics in the natural sciences."
- Specialization: Statistics, computer science, and operations research are independent disciplines.
- Aesthetics: The appreciation of elegance, beauty, and simplicity in proofs.
- Aesthetic examples: Infinite prime numbers (Euclid) and the Fast Fourier Transform.
- Notation: Most current symbology dates back to the 18th century (Euler).
- Language: The use of technical jargon (homeomorphism, integrability) to achieve precision.
- Rigor: An indispensable condition of systematic reasoning starting from axioms.
- Evolution of rigor: Variation from antiquity to 19th-century analysis and computer-assisted research.
- Axiom: A chain of symbols with meaning only within an axiomatic system.
Nature, classification, and awards in mathematics
- Definition of science: field of knowledge
- Mathematical theories: hypothetico-deductive in nature (Popper)
- Computational mathematics: proposed as a scientific field in 2002
- Fields Medal: awarded every 4 years since 1936
- Wolf Prize: created in 1978 for lifetime achievements
- Abel Prize: created in 2003 for innovative research
- Hilbert's Problems: 23 problems (1900), 9 solved
- Millennium Prize Problems: 7 problems (2000), $1 million reward
- The only problem on both lists: Riemann Hypothesis
- 5 basic areas: Quantity, Structure, Space, Change, and Variability
- Applied Mathematics: "outward" development approach
- Pure Mathematics: "inward" development approach
Statistics and Decision Sciences
- Definition: study of variability and random processes through laws of probability
- Application: fundamental in technology (computer science, engineering) and factual sciences (economics, genetics, sociology, psychology, medicine, accounting)
- Methodology: requires collection, organization, presentation, and analysis of numerical data
- Scientific objective: obtaining knowledge through experimentation, observation, and inferences under a theoretical framework
- Universal purpose: making intelligent decisions under conditions of uncertainty
- Descriptive statistics: basic level of instruction with simplified concepts and techniques
Flashcard preview (6 of 25)
- 01
Etymology: From which Greek word does 'mathematics' (*mathēmatiká*) derive?
*máthēma* (that which is understood)
- 02
What type of science is mathematics according to its formal definition?
Formal science
- 03
What concepts are considered objects of study in mathematics?
Quantity, shape, space, and number
- 04
Who developed the concept of a group to describe the symmetries of an object?
Niels Henrik Abel and Évariste Galois
- 05
Who published 'The Foundations of the Theory of Probability' in 1933?
Andrei Kolmogorov
- 06
Etymology: What does the Greek term *mathēma* (μάθημα) mean?
Field of study or instruction.
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