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. Numbers, infinity sign ○ Numbers and the rules with which they are expressed can vary ○ But they ultimately mean the same thing ○ E.g. 243 vs “ 两百四十三 ” (Chinese) ■ Place assumed in English vs Placeholders such as “ 百 ” (hundred) are employed in Chinese The use of idealisation and abstraction results in foundational truths known as AXIOMS which kickstart mathematical knowledge construction Axioms ● Self-evident truths ​ taken to be established that serve as the ​ foundation for mathematical knowledge ○ Also known as postulates ○ //General Epistemology: Foundationalism ■ Difference: ​ Analytically/Pragmatically ​ rather than dogmatically derived ● In contrast with religious knowledge construction where most start from dogmatic foundations about the nature of God to deduce further truths (e.g. God is always right → God’s law is always right),

mathematical axioms are self-evident as they are analytic (true by the nature of their definition rather than some external truth e.g. All bachelors are unmarried) ○ Deductions are made from Axioms to form an Axiomatic-Deductive System ■ //Intuition-Deduction Thesis in Rationalism ● Example: Euclid’s axioms ○ Two things equal to a third are equal to each other ■ x = y, z = y, x = z ○ Equals (even numbers) added to equals make equals ● Self-evident nature seems to imply both realism and anti-realism ○ Anti-Realist Interpretation: Axioms are invented to suit what the concepts are intended to reveal about reality ○ Realist Interpretation: Axioms are known through Platonic Ideals/Aristotelian Intuition ● Sometimes axioms are established whilst the branches of postulates are being added ○ I.e. do not have to dogmatically start from axioms; logic and practical use also advise Criteria for Axioms 1. Consistent: Implications ​ should not contradict a. Should not be possible to derive P and ~ P (not-P) from axioms b. Principle of Explosion: One contradictory axiom would lead to further contradictory theorems thus ruining the system c. // Construction method of Proof d. // Coherentism 2. Independent: Should not be possible to derive one axiom from the others 3. Fruitful: Should be possible to derive many theorems from axioms ● Suggests that axioms are defined by utility in consistency ○ Try, for example, to make a new number system that's like the ordinary one except that it skips some number -- say, 4. It just won't work. Everything will go wrong. You'll have to decide what 2 plus 2 is. If you say that this is 5, then 5 will have to be an even number, and so also must 7 and 9. Then, what's 5 plus 5? Is it 8, or 9, or 10? You'll find that to make the new system at all like arithmetic you'll have to change the properties of all the other numbers. Then, when you're done, you'll find that you have changed only those numbers' names and not their properties at all. Similarly, you could try to make two different numbers be the same -- say, 139 and 145. But then, to make subtraction work, you'll have to make 6 the same as 0 and 4 plus 5 equal to 3. Suddenly, you'll find that the sum of two positive numbers is smaller than either of them--and that scarcely resembles arithmetic at all.

Sources of Axioms ● Analytically Deduced Axioms: Some concepts have their meaning and properties derived entirely from abstract axioms that prescribe their properties ○ E.g points and straight lines as opposed to triangles ○ A Priori ○ Cognitively Pragmatic ​ Axioms: Axioms that are defined as such due to their usefulness in our thinking ■ E.g. Multiplication: a x b x c may be (a x b) x c, (a x c) x b or even (a simultaneous application of multiplication to all three numbers ■ However, we accept the first two definitions of multiplication (and its permutations that multiply two numbers at a time) and do not accept the simultaneous application to all three numbers ■ Such a characteristic within the definition of multiplication would demand the need for a new definition that accommodates 4 numbers, 5 numbers, 6 numbers and so forth ■ //Kant on constructivism: Are these axioms defined as so because of the way the world is, or because of the way our mind constructs the world? ● Physical Knowledge Axioms: Some concepts have their meaning and properties derived from an ​ application in Science ○ Mathematics serves as the language of Science ■ Density as mass divided by volume ■ Force as the product of mass and acceleration ○ Imposes Arbitrary quantities on natural phenomena ■ Paradox: The ‘invented’ nature of mathematics vs its strong relevance to the real world ○ //Abstraction: Beneficial for clarity in constructing scientific knowledge in an objective manner Deduction ● //General Epistemology: Reasoning that moves from the general to the specific ● Suggests that no new propositional knowledge is gained throu