Math as a Way of Knowing Theory of Knowledge Nature and Methodologies of Math as an AOK Individual Questions Universal Math Invented or Found? Mathematical Proof The Math Process Is there a fundamental difference in the approach to mathematical knowledge between pure and applied math How math Evolves Nature of Math Knowledge Claims Knowledge Issues Knowledge Claims and Knowledge Issues -Since math is deductive using axioms as the given starting foundation for all mathematical knowledge (theorems, equations etc...), to begin exploring math as an area of knowledge, one must accept the requirement for a congruent set of axioms. -Yes, because knowledge in pure math is only supported by axioms but by empirical evidence in applied math

-These axioms are not proven to be knowledge so pure math in a way is not even knowledge

-Pure math is deductive while applied math is inductive

-Pure math is like the plot of a fictional work while applied math is like a non-fictional work

-Pure math explores the boundaries of reason like a fictional work explores boundaries of imagination

-Applied math simply reflects reality Math needs premises to start of deducting reasoning

Have to start somewhere As discussed earlier there is not absolute math.

But, there is a chance for it to be universal... What do mathematicians mean by mathematical proof, and how does it differ from good reasons in other areas of knowledge These premises (axioms) allow mathematicians to

discover "new" math

"New" Math needs to be proven as true

If the axiom is true, the result is true Establish an axiom as

irrefutably true Deduct based on that axiom Find an answer Determine its validity,

possibly establish a new theorem Pure Math vs. Applied Math -Math is deductive

- "a priori" (Knowledge comes before experience)

-Uses axioms as premises

-Axioms are 'facts' that are assumed to be true Example -Axiom: a+b=b+a

-Deduction:0. 5+2=2+5 Math is a Rational and Axiomatic System Certainty in Math -There is no absolute math

-A different set of axioms form a different math

-All knowledge in math is certain based on the axioms To prove your discovery to be true, one has

demonstrate that it is always true in all scenarios Euclidean Geometry

-Sum of angles in triangle = 180 Saddle Geometry

-Sum of angles in triangle < 180 Spherical Geometry

-Sum of angles in triangle > 180 Math was created by humans to quantify the world, though the foundations of math we discovered Did we create math or did we find it? There can be one of a single object, but there is no such thing as a "one" in nature. How do we know if certain evidence approves of or disproves of (applied) math? Applied math is composed of theorems and laws that are meant to represent real world relations and patterns. -As math was created out of the human need to quantify, and later complete more elaborate tasks, one must also accept the concept of math applying to the real world.

-If one only works with pure math, then the most that can arguably be gained is justified belief that isn't necessarily true.

-When one enters the area of applied math Who defines the theoretical boundaries and limitations of mathematics? To what extent do the variations between mathematical systems make math less absolute? If math is abstract on its own, how reliable is it to obtain knowledge in the real world? If math is centered around perfection, to what extent does math mirror the real world? Showing that if the axiom you acknowledged as absolute is true, then the answer you got based on that axiom is true. If you can successfully demonstrate it, your answer is a theorem, and other mathematicians can build on it. What is Mathematical Proof How to prove it? Logically combining axioms and previous theorems

if... then...

if not... then not...

Specific Scenario

Probability Whereas other AoKs combine different ways of knowing to find new knowledge, Mathematics uses only reason Reason and Deduction as opposed to Emotion and "Gut Feeling" Pure Math Applied Math Can math be characterized as a universal language? -Rules are arbitrary but allow for safe driving

-A good driver always follows rules -Set of axioms being used are arbitrary

-Good mathematician always agrees with axioms -Relies on axioms

-No real life application or representation

-3*2=6 -Models real life phenomenons

-Ties in with natural sciences

-PV=nRT There are many instances where math fails to articulately represent real world relations. In such cases, some might argue it is impossible for real world evidence to approve math, and vice versa. The theoretical and practical definitions of math do not always match up.

There are mathematical expressions in theory that cannot be evaluated, and there are expressions that can be evaluated but are undefined in theoretical mathematics. Any new additions to the mathematical system require evaluation regarding whether they is doable (i.e. practical) and whether they can co-exist with existing theoretical mathematical rules There are multiple mathematical systems, with varying methodologies and rules, but all of them agree on the same axioms. The use cases of mathematical systems often overlap, resulting in varying methodologies for a single mathematical problem. This could make math less absolute if the solutions vary, which violates its nature of being absolute. What is the role of "beauty" and "elegance" in mathematical thought? Mathematical representations (e.g. numbers) can be evaluated based on beauty and/or elegance. In fact, the whole concept of simplification in math is based on elegance. For example, 4096/1024 is less elegant than 4, while both numbers represent the same value. Mathematical methodologies and theorems can also be evaluated based on elegance, but this is a secondary factor and is subordinate to aspects like accuracy and precision. Commutative Laws:

The commutative laws state that the order in addition and multiplication does not matter. Thus x+y=y+x and xy = yx. Associative Laws:

The associative laws states that in repeated multiplication or addition, grouping does not matter. We can look at an example of the associative laws:

x+(y+z) = (x+y) +z = x+y+z Distributive Laws:

The distributive laws states that multiplication is, as the axiom implies, distributed over addition. An example of the Distribute Laws is:

x(y+z) = xy+yz Identity Laws

From the identity laws we get two things.

First, there is a unique number 0 with the property that 0 plus any given number equals the number itself. 0+x = x Inverse Laws

The last axiom of operations is the inverse laws. The inverse laws state first that for any real number, there is a negative of that number (the additive inverse), that when added together equals zero. x + (-x) = (-x)+x = 0

The inverse laws also states that for any real number except 0, there is the same number to the power of -1 (the multiplicative inverse, or reciprocal), that when multiplied equals. x × x^(-1)= 1

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