RI Math Lecture 1 2023

Mathematics Lecture 1: Introduction and the Usual Suspects

OverviewuEssential Questions uPreliminaries/ DefinitionsuThe Mathematical Method vs the Scientific MethoduA Priori vs A Posteriori?uThe Usual Suspects: Empiricism vs Platonism

Essential QuestionsuHow do we construct Mathematical Truths? Reason or Experience? Inductive or Deductive reasoning?uAre Mathematical Truths Created or Discovered?uDo such Truths exist in the Physical Realm or a Mental Realm?uHow certain are Mathematical Truths? Absolutely or relatively?uAre Mathematical Truths universal, holding true for all times and place? Or only relative? i.e. Objective vs Subjective?uIs Mathematics useful?uHow is Mathematics different from other areas of knowledge? uIs it more certain? (e.g. Social Science)uIs it more fundamental? (e.g. Science)uIs it more objective/universal? (e.g. Ethics)uIs it more justified? (e.g. Aesthetics)uIs it more useful (E.g. History)

Preliminaries/ DefinitionsuAxioms: assumedstatements in a theory; no proof is givenuTheorem: statements in a theory that are implied by the axiomsuTheory: a set of sentences expressed in a formal languageuFormal language: a language two of whose features are formally specified: the linguistic symbols of the language and rules for joining together or concatenating these symbols into well-formed formulae or words which can be assigned precise meanings. (Oxford Dictionary of Philosophy) uEssentially, a formal language is the symbolic version of the normal, everyday language that we use.uE.g. The sentence “For all X that belongs to the set of natural no.” can be translated to something like this

E.g.: Euclid’s axiomsu1. It shall be possible to draw a straight line joining any two points.u2. A finite straight line may be extended without limit in either direction.u3. It shall be possible to draw a circle with a given centre and through a given point.u4. All right angles are equal to one another. u5. There is just one straight line through a given point which is parallel to a given line.

E.g.: Euclid’s theoremsu1. Lines perpendicular to the same line are parallel.u2. Two straight lines do not enclose an area.u3. The sum of the angles of a triangle is 180 degrees.u4. The angles on a straight line sum to 180 degrees.

HDM vs ATM

Axiom-Theorem nature of MathMethodExample1.Start with Question/ problemDo two odd numbers added together always give an even number? 3. Theorem: Through deductive reasoning, find a proof to solve the problem.Let the odd numbers be a and bThen a = 2n+1b = 2m+1 for whole numbers n and ma+b= 2n+1+2m+1= 2m+2n+2= 2(m+n+1)= 2p where p is whole no.But this is of the form 2n and hence even (QED) 2. Axiom: Start with assumptions and definitions; check these definitions.Even number = 2nOdd number = 2n + 1, where n is whole numberLaws of arithmetic apply

Hypothetico-Deductive Method in ScienceMethod Example1.Science begins with an interesting observation Diet Coke (DC) floats! 2. Observation leads to a question ‘Why does DC float while regular coke sinks?’ 3. Question leads to hypotheses (via inductive reasoning) which allows us to make a prediction (using deductive reasoning). ‘DC floats because aspartame is less dense than sugar’‘Aspartame + water will also be lower density than sugar + water’4. Experiments are conducted to test the hypotheses: If the hypothesis is correct, then prediction will be verified by the experiment.If the hypothesis is wrong, then it’s eliminated and a new hypothesis composed. All other variables constant, it is the presence of aspartame in place of sugar that causes the diet coke to float.‘DC floats because the DC cans are not filled to the brim’ 5. After repeated tests, a hypothesis is corroborated and becomes scientific fact / theory.Aspartame is of lower density than sugar.

Inductive or Deductive?uMight seem a little duh to ask this question since Mathematics seems to be the one area of knowledge which is totallydeductive in nature.uFor example, 2+2=4 seems a deductive entreprise. We don’t go out there, take 2 items, and another 2 items, and then count that there is 4 and then from there generalize it to all items. We couldbut we don’t have to.uWhy?