Scope & Application
Mathematics is the study of...?
Patterns
Commutative property, reducing fractions, and cross multiplication are all examples of _____
mathematical concepts.
Which type of reasoning is math heavily dependent on?
Deductive Reasoning
How long did it take to prove Fermat’s Last Theorem
356 years
What is the name of the strategy used by Cantor in his explanation of the existence of multiple infinites?
reductio ad absurdum
State 2 fields that are included in pure mathematics.
Algebra, analysis, geometry, number theory, and topology
What is the function of the symbolic system?
Allows ideas to be manipulated in mind, and then shared with others.
What is the fallacy associated with deductive reasoning in this AOK?
False premises
What did Kurt Godel publish?
Kurt Godels Incompleteness Theorem, “Having Mathematics reach a state of completeness is impossible”
what are the 3 schools of thought regarding the existence of numbers?
nominalism, platonism, fictionalism
State 4 fields that are included in applied mathematics.
numerical analysis, scientific computing, mathematical physics, information theory, control theory, and actuarial science
Name 2 characteristics of the symbolic system (from the presentation)
Precise & compact
Name or describe the two main methods discussed in this presentation
1. Building on Foundations - deductive reasoning 2. Proof and Peer review
Who proved Fermat’s Last Theorem, and how long did it take him?
Andrew Wiles, 8 years
Collaboration is used to determine: ________?
Validity
What is the concept of platonism?
Numbers exist but are abstract and they exist outside of space and time.
Discuss the relationship between language (words) and symbols (numerics) in mathematics?
It would be impossible to develop shared knowledge about numerics and numerical concepts if it weren’t for language.
What common metaphor is used for axioms in developing mathematical theories/claims?
The axioms are the base of the building that new claims are constructed upon (a strong base makes for a strong building)
What are Alan Bishop’s 6 factors?
Counting, Locating, Measuring, Designing, Playing, and Explaining
Imagination refers to?
The creative capacity to reassemble familiar components into new ones or to project beyond them into fresh conceptualizations
True or False:
Nominalism states that numbers do not exist and mathematical discourse is systematically false despite being useful. It will try to explain things without saying mathematical claims are true.
False, that is fictionism.
How are mathematical concepts defined?
“Deductive theory”: by the means of primary notions, other notions are defined.
Which Euclidean axiom/postulate was disregarded in Riemann’s Geometry?
The fifth axiom: The Parallel Postulate
What is Russell’s Paradox
If he created a set of all sets that are not members of themselves, it created a contradiction