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*Math is created with logic deduction-based,what about a degree of induction?(INVENT-TEST/OBSERVE-CONSTRUCT-TEST) (invention Vs discovery)-The role of empirical evidence,needs formula to comprehend? Is formula derived fromevidence too? Circular?-Modus Ponens : If P, then Q. P, therefore, Q.-Modus Tollens :

*People express their understanding of some aspect of the world through mathematicalformulations.

*The NAMING,CONVENTION,IMPOSITION, of mathematical claims are individual-dependentand subjective. These are INVENTED constructs.

*But is there an objective truth we need to seek out? And is this through discovery alone?

*Epistemic IssueHow do we know?does it matter?

*AXIOMS: definitional statements/assumptions/definitions/taken without ques tion cannot beproven.(Invented not Discovered)

-Starting points from which consequences inevitably follow.

-Need to know parameters-often assumption-based: follows a logical-sequence?-Must corroborate, consistent NOT conflicting with other conditions .-Axiom must be consistent with usage.-Working axiom is assumed to be accurate,usage on relative perceived accuracy.-Axiom+ logic = Theorem-A given set of axioms gives a specific set of results in separate contexts.-How did these come about?-Changeable subject to new insights and ideas. (certainty compromised?)-Axioms-----(logical deduction,proof)-Theorem-How did the axioms come about? ± were they discovered [objective existence] ORinvented [human, social, cultural, political, « agendas]

 Academic Mathematics Students explore the basis of mathematical knowledge obtainedthrough deductive reasoning and building from clearly stated assumptions and arriving atsystematic conclusions through a process of rigorous logic.

Math And Truth (assumed to be constant) {TRUTH/PROVEABLE}-Statements can be proven,but is that the truth?-Consistency: A mathematical truth corresponds to reality-If all statements are logically proven,but not consistent (given anomalies NOT GOOD!)-True statements that are NOT PROVABLE exist.An axiom is complete if every true statement

is provable.-There are TRUE things we can never know by proving but there is confidence that method isinevitably right.-Math + deductive logic = Conferring certitude-Consistency = Truth specific wordsimply attitude.

*Math is never empirical UNLIKE science.Conclusions may lose value due to falliable axioms but method of proving is accurate.

*Math is to create a further generalization/abstraction of our world?

-Hilbert,Godel,Paul Ernest.Socrates,B.F Skinner,Euclid

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 -Freeman Dyson Math as a fundamental science used to generate hypothesis about thenworld. (Construction of Knowledge)

-Martin Gardener Mathematics as the absolute.Perceiving the world through mathematicalconventions/structure.

-Paul R Halmos Mathematics similarities to science deductive formulation,absolute answer.

-Godfrey H Hardy mathematics has an aesthetic value? Beauty in math or banalcalculations? The mathematical arena as a basis for redefining beauty. (sensi tive to the use of the term).Harmonizes several disciplines. Math is a reality that exists to address the beauty of our externality.

-Emmanuel Kant: Mathematics as a solely deductive agency (schema of math) noobservation, no real life data. Rationalist stance.

-J.E. Littlewood :Mathematics corresponds to the real world (truth?) But mathematics is notmeant to be tested,the conventions in math make up a separate/parallel reality?

-James Roy Newman : Mathematics predictive abilities have a relation to the physical world

but mathematics exists independent of the ³real world´. Can enable us to ³know´. Formulation----- Physical reality.

-Paul Ernest : ³According to social constructivism, mathematics is more than a collection of subjective beliefs, but less than a body of absolute objective knowledge, floating above allhuman activity. Instead it occupies an intermediate position. Mathematics is culturalknowledge, like the rest of human knowledge.´

-From axioms we can use the rules of logic to derive what logically follows and to find other results which are called theorems.Theorems: statements we logically deduce from theaxioms.

Theorems are known with complete certaintyProof is important

European Mathematical Society Newsletter - June 2008 Issue 68Pg 19, ³On Platonism´- Reuben HershUniversity Of New Mexico,USA

-Platonism vs Formalism

*Platonism :-mathematical objects exist independently of our knowledge or activity

-mathematical truth is objective [same status as scientific truth]-mathematical entities are ³out there.´

- recognizes that mathematical facts and entities are not subject to the will or whim of theindividual mathematician-has objective facts and entities whose independent exist ence and qualities seek recognitionand discovery.

Incompatibility- with the standard view of the nature of reality (physical world). Nature of mathematical truth and existence is to be understood in that realm (physical reality), rather than in some other independent ³abstract´ reality.

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-The fallacy of Platonism is in the misinterpretation of objective reality, putting it outside of human culture and consciousness. Like other cultural realities, it is external, objective,from theviewpoint of any individual, but internal, historical, socially conditioned, from the viewpoint of the society or the culture as a whole.

*Formalism-math is logical deduction from formally stated ³axioms,´-³facts´ about mathematical objects have no ³meaning´ or ³reference,´ just logicalmanipulation of logical formulas.

Incompatibility- with the actual practice of mathematics, whether in research or in teaching.

 Axioms : certain assumptions and definitions which are taken without question and wh ich

cannot be proved [1+1=2] -Axioms are important in Mathematics and are statements which we accept without questionand agree to be true.-Context to which we apply axioms can differ thereby leading to different µanswers¶-Mathematical axioms can change ± Euclidian plane geometry giving way to other geometrieswith different axioms-From axioms we can use the rules of logic to derive what logically follows and to find other results which are called theorems ± cf DAB

-Theorems are known with complete certainty-Proof is important-how did the axioms come about? ± were they discovered [=> objective existence] ORinvented [=> human, social, cultural, political, « agendas]-mathematical concepts can change with new ideas/insights [e.g., geometry]-Mathematicians are not bothered about the truth of axioms (truth=> correspondence toµreality¶) 

-Distinctions:-Mathematics as a body of knowledge (Pythagoras, Fermat, calculus, number theory, etc)VsPractice of doing Mathematics (proofs, books about maths, etc)-Statements which are true vs statements which are provable

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