British philosophy of mathematics
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This document discusses the British philosophy of mathematics. It contains a quote from A.J. Ayer's "Language, Truth and Logic" that empiricists must either say logical and mathematical truths are not necessary or that they have no factual content. The document then lists notable British mathematicians from the 15th-17th centuries including Cardano, Viète, Harriot, Pell, and Torricelli.
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British philosophy of mathematics
This document discusses the British philosophy of mathematics. It contains a quote from A.J. Ayer's "Language, Truth and Logic" that presents empiricists with two options for explaining the necessity of logical and mathematical truths: either they are not truly necessary, requiring an explanation for their universal acceptance as such, or they have no factual content, requiring an explanation for how non-factual propositions can be true, useful, and surprising. The document also lists some key figures in the development of mathematics between 1545-1668.
computer
Mathematics is the abstract study of topics such as quantity, structure, space and change. Mathematicians seek patterns and use them to formulate conjectures, which they then aim to prove or disprove through mathematical proof. Practical mathematics has been a human activity for as far back as written records exist, while rigorous arguments in mathematics first appeared in Greek mathematics. Today, mathematics is used throughout the world as an essential tool in many fields, and both pure and applied mathematics continue to develop and inspire new discoveries.
Euclid
Euclid was a Greek mathematician who lived in Alexandria, Egypt around 300 BC. He is considered the "Father of Geometry" and authored the influential textbook Elements, which systematically outlined the principles of geometry and mathematical reasoning. The Elements was used as the primary textbook on mathematics for over 2000 years. It began with definitions, postulates, and axioms and proceeded to prove 467 theorems through logical steps and deductive reasoning. Euclid used a "synthetic" approach to present his theorems by progressing logically from known facts to unknown conclusions. The Elements established geometry as a logical science and influenced mathematics education for centuries.
Mathematics
Mathematics is the study of topics such as quantity, structure, space, and change. Mathematicians seek patterns and use them to formulate conjectures, which they then aim to prove or disprove through mathematical proof. When mathematical structures accurately model real phenomena, mathematics can provide insight and predictions about nature. Mathematics has developed through concepts like counting, calculation, measurement, and the study of shapes and motions, and is used throughout the world as an essential tool across many fields like science, engineering, medicine, and finance.
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The document discusses the concept of mathematical proof and debates around its nature. It argues that viewing proof as a Wittgensteinian family resemblance concept or cluster concept is insufficient, as mathematics is not just a collection of discrete language games. Rather, we have a single rich incoherent concept of proof. This presents difficulties for those who want to make ideological use of the concept or teach it. The document also examines statements from the "standard view" that proofs should be formal, and resistances to this position, for example that informal proofs provide understanding in a way formal proofs do not.
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Lakatos demonstrates the possibility of philosophical hybridism by combining influences from Hegel, Pólya and Popper in his philosophy of mathematics, which seems unlikely but points to the eclectic nature of the philosophy of mathematical practice. The talk will discuss the tensions and synergies in Lakatos’s thought from his intellectual influences and migration, and illustrate the benefits of philosophical hybridism, including a robustness criterion. It will also discuss prospects for collaboration among various projects under the philosophy of mathematical practices.
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British philosophy of mathematics
This document discusses the British philosophy of mathematics. It contains a quote from A.J. Ayer's "Language, Truth and Logic" that presents empiricists with two options for explaining the necessity of logical and mathematical truths: either they are not truly necessary, requiring an explanation for their universal acceptance as such, or they have no factual content, requiring an explanation for how non-factual propositions can be true, useful, and surprising. The document also lists some key figures in the development of mathematics between 1545-1668.
computer
Mathematics is the abstract study of topics such as quantity, structure, space and change. Mathematicians seek patterns and use them to formulate conjectures, which they then aim to prove or disprove through mathematical proof. Practical mathematics has been a human activity for as far back as written records exist, while rigorous arguments in mathematics first appeared in Greek mathematics. Today, mathematics is used throughout the world as an essential tool in many fields, and both pure and applied mathematics continue to develop and inspire new discoveries.
Euclid
Euclid was a Greek mathematician who lived in Alexandria, Egypt around 300 BC. He is considered the "Father of Geometry" and authored the influential textbook Elements, which systematically outlined the principles of geometry and mathematical reasoning. The Elements was used as the primary textbook on mathematics for over 2000 years. It began with definitions, postulates, and axioms and proceeded to prove 467 theorems through logical steps and deductive reasoning. Euclid used a "synthetic" approach to present his theorems by progressing logically from known facts to unknown conclusions. The Elements established geometry as a logical science and influenced mathematics education for centuries.
Mathematics
Mathematics is the study of topics such as quantity, structure, space, and change. Mathematicians seek patterns and use them to formulate conjectures, which they then aim to prove or disprove through mathematical proof. When mathematical structures accurately model real phenomena, mathematics can provide insight and predictions about nature. Mathematics has developed through concepts like counting, calculation, measurement, and the study of shapes and motions, and is used throughout the world as an essential tool across many fields like science, engineering, medicine, and finance.
The concept of proof: how much trouble are we in?
The document discusses the concept of mathematical proof and debates around its nature. It argues that viewing proof as a Wittgensteinian family resemblance concept or cluster concept is insufficient, as mathematics is not just a collection of discrete language games. Rather, we have a single rich incoherent concept of proof. This presents difficulties for those who want to make ideological use of the concept or teach it. The document also examines statements from the "standard view" that proofs should be formal, and resistances to this position, for example that informal proofs provide understanding in a way formal proofs do not.
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Lakatos demonstrates the possibility of philosophical hybridism by combining influences from Hegel, Pólya and Popper in his philosophy of mathematics, which seems unlikely but points to the eclectic nature of the philosophy of mathematical practice. The talk will discuss the tensions and synergies in Lakatos’s thought from his intellectual influences and migration, and illustrate the benefits of philosophical hybridism, including a robustness criterion. It will also discuss prospects for collaboration among various projects under the philosophy of mathematical practices.
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2) Logic should be understood as the general study of inferential actions, as informal proofs involve actions on mathematical objects beyond just propositions.
3) Representations in mathematics are valuable not just for presenting information but for being manipulated, as inferences can be drawn through performing actions on diagrams, graphs, and other representations.
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