1. MATHEMATICS

2. https://youtu.be/zAxT0mRGuoY

3. Bellwork
Can you imagine a world in which 2+2=5? For example, what if ever time you brought two pairs of objects close to one another, a fifth one popped into existence?

4. Mathematics and certainty
Analytical: True by definition
Synthetic: Not Analytic
A Priori: True Independent Experience
A Posteriori: Not a true independent experience

5. Box 1: All definitions in this box Box 2: Empty
Distinction matrix
Box 1: All definitions in this box
Box 2: Empty
Box 3: Empirical Knowledge
Box 4: Unkown
Analytic
Synthetic
A priori
Yes ?
A posteriori No

6. Empirical Mathematics
“Mathematical Truths are empirical generalizations based on a vast number of experiences”
Common knowledge being 2+2=4, but not necessarily that metal expands when heated.

7. Discussion
Question 1 on Page 199

8. Analytical Mathematics
Truth is already present in mathematics, the “wrapper” simply has to be taken off
Good short-term memory is important to mathematics
Goldbach’s Conjecture almost contradicts analytical mathematics

9. MATHEMATICS AS SYNTHETIC A PRIORI
It could also be argued that mathematics fits neither the empirical or analytical boxes, but in synthetic a prior knowledge.
Reason must be used to arrive at natural truths
“To speak freely, I am convinced that it
“To speak freely, I am convinced that it [mathematics] is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others” - Rene Descartes

10. Mathematical truths are empirical generalizations
The three views of math
Empiricism
Box 3
Mathematical truths are empirical generalizations
Formalism
Box 1
Mathematical truths are true by definition
Platonism
Box 4
Mathematical truths give us a priori insight into the structure of reality

11. Discovered or invented?
Platonists believe that mathematical entities are discovered
Formalists argue that they are invented and only exist in the mind

12. Group Discussion
Discuss how you would explain mathematics as being both invented and discovered

13. Mathematics is more certain than perception
Plato’s argument
Mathematics is more certain than perception
Mathematics is timelessly true
Two objections to Plato
If there are an infinite number of math entities, it is hard to prove or argue the observation of infinity
If the entities have some ideal existence naturally, how do humans learn them?

14. Differed from Euclid’s Two points may determine more than one line
Riemann’s Axioms
Differed from Euclid’s
Two points may determine more than one line
All lines are finite in length but endless - i.e. circles
There are no parallel lines
Theorems Derived From Above Axioms
All perpendiculars to a straight line meet at one point
Two straight lines enclose an area
The sum of the angles of any triangle is greater than 180 degrees

16. Problem of consistency
Riemann believed that because Euclid’s axioms ran into contradictions, his would have contradictions as well
However, intuition alone cannot determine contradiction
Gödel’s Incompleteness Theorem
Didn’t discover any contradictions
Stated that math isn’t certain for we may one day discover a contradiction

17. Math only serves a purpose when it is applicable to the real world
Applied Mathematics
Math only serves a purpose when it is applicable to the real world
An example is Appalonius studying ellipses in Greek times. Ellipses were useless to them at the time, yet it was still a mathematical discovery
Einstein says that mathematics is invented, but it is a matter of discovery which of the various systems apply to reality.
Buffon’s Needle Problem