Maths__Luosuo_Version___1_

Knowledge and Inquiry Questions / ​ Parallels ​ / ​ Main Ideas Notes Maths: Its Nature, Construction and Validity Readings I plan to use 1. Was Mathematics Invented or Discovered: Realism vs Anti-realism 2. Reason_and_arithmetic_Charles_Landesman: Mathematics in philosophy (12) 3. Mathematics_as_an_objective_science_by_Nicolas_D._Goodman: (13) 4. Mathematics-Cultural_product_or_epistemic_exception__by_Susanne_Prediger Readings Used 1. What is Mathematics 2. The Rise and Role of Mathematics: Mathematics in philosophy 3. The Three Crises in Mathematics: Different Schools of thought 4. The Unreasonable Effectiveness of Mathematics in the Natural Sciences 5. Beliefs Shape Mathematics (3) 6. Is Mathematics Philosophically Analytical (3) 7. Revisiting the 'unreasonable effectiveness' of mathematics (9) Mathematical Propositions have some distinct characteristics such as... Nature of Mathematics Possesses ​ Basic Concepts ​ based on: ● Material/Physical Concepts: E.g Geometry: Point, Line, Plane ● From Ideas: E.g. Negative numbers, calculus, complex numbers ○ ≠ Science and Art ● A combination of both: E.g. Instantaneous rate of change Abstraction Definition: Process of ​ extracting ​ the underlying essence of a mathematical concept ● Removal of any ​ dependence on real world objects ​ with which it might originally have been connected ● Does not occupy spatial and temporal locations; neither does it engage in causal relationships with other objects. ● Generalization ​ so that it has wider applications or matching among other abstract descriptions of equivalent phenomena Example: Buying 3 pairs of shoes for $20 ● Abstract 3 and 20 and multiply to obtain 60 ● Interpret the result to suit the physical situation

Concrete Real World Concept → ​ Abstraction ​ → Mathematical abstract structure ● E.g. Geometry: Calculation of distances and areas in the real world → Properties of shapes ● One abstract concept should be applicable to ​ all the physical manifestations ​ of that concept ● Ongoing Process ○ E.g. Geometry: Euclid's Elements + Cartesian Coordinates ● Suggests that Mathematics began as a practical tool that was abstracted to fit more applications Advantages: Heavy Synergy and Connectedness ● Reveals deep connections between different areas of mathematics ○ Unity of algebra and geometry with cartesian coordinates ○ //Positive Coherentism ● Known results in one area can ​ suggest conjectures ​ in a related area ● Techniques and methods from one area can be applied to prove results in a related area Idealisation Definition: The assumption that any concept is considered as its ideal ● In applying mathematics, we consider the ​ ideals of objects ​ rather than their imperfect physical forms ● E.g. Regarding the Earth as a perfect sphere ● Application of Abstraction to the physical world Use of Symbols ● E.g. Num