The Rising Sea
Writing Mathematics
Mridul Biswas
Mathematical idea outlives any language
Pythagoras
(c. 570 – c. 495 BC)
Mathematical idea outlives any language
Pythagoras
(c. 570 – c. 495 BC)
Wanderer above the Sea of Fog
(Caspar David Friedrich)
Intuition and Rigor
Intuition and Rigor
Wanderer above the Sea of Fog
(Caspar David Friedrich)
Intuition shows the horizon. Rigor gets you there.
Intuition and Rigor
Wanderer above the Sea of Fog
(Caspar David Friedrich)
Intuition shows the horizon. Rigor gets you there.
Breaking Down, Building Up
Breaking Down, Building Up
Start
Start
Statement
Start
Statement
Understand defs
And try examples
(by hand)
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
False Statement
Yes
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
Mimic example
False Statement
Yes
No
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
Mimic example
False Statement
Proof
Yes
No
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
Mimic example
End
False Statement
Proof
Yes
No
Mimic example
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
End
False Statement
Proof
Yes
No
Start
Statement
Understand defs
And try examples
(by hand)
Counterexample?
Mimic example
End
False Statement
Proof
Yes
No
The sum of two even numbers is even.
What Makes a Method Powerful?
What Makes a Method Powerful?
Exploration
Opens paths to related problems
What Makes a Method Powerful?
What Makes a Method Powerful?
Counterexample
Reveals natural counterexamples
Domain Flexibility
Structural idea, not domain-specific tricks.
What Makes a Method Powerful?
What Makes a Method Powerful?
Breaking Down, Building Up
Style: Explore examples, mimic patterns, try to generalize.
EX
CE
IN
FP
DF
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
Sandwich Method
Style: Start from both statement and goal, meet in the middle.
EX
CE
IN
FP
DF
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
Zoom-Out Method
Style: Step back, see global structure, explore big patterns
EX
CE
IN
FP
DF
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
Framework Mimicry
Style: Mimic a known proof for a similar problem
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
EX
CE
IN
FP
DF
Blackbox Method
Style: Assume statement, connect to known
EX
CE
IN
FP
DF
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
EX
CE
IN
FP
DF
EX
CE
IN
FP
DF
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
EX
CE
IN
FP
DF
Breaking Down, Building Up
EX
CE
IN
FP
DF
Sandwich Method
EX
CE
IN
FP
DF
Zoom-Out Method
Framework Mimicry
Blackbox Method
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
EX
CE
IN
FP
DF
Breaking Down, Building Up
EX
CE
IN
FP
DF
Framework Mimicry
High School Problems
EX: Exploration
IN: Intuition
CE: Counterexample
DF: Domain Flexibility
FP: Finishing Proof
EX
CE
IN
FP
DF
Breaking Down, Building Up
EX
CE
IN
FP
DF
Framework Mimicry
EX
CE
IN
FP
DF
Sandwich Method
Undergraduate Level Problems
Crack the Nut, Don’t Become One
Hammer
Crack the Nut, Don’t Become One
Hammer
Water
Crack the Nut, Don’t Become One
Are most square matrix invertible?
Are most square matrix invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
Are most square matrix invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
Are most square matrices invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
Are most square matrices invertible?
Are most square matrices invertible?
Are most square matrices invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
Are most square matrices invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
Are most square matrices invertible?
Are most square matrices invertible?
Are most square matrices invertible?
Are most square matrices invertible?
Are most square matrices invertible?
EX
CE
IN
FP
DF
Framework Mimicry
Are most square matrices invertible?
EX
CE
IN
FP
DF
Breaking Down, Building Up
EX
CE
IN
FP
DF
Framework Mimicry
Are most square matrices invertible?
Are most square matrices invertible?
Are most square matrices invertible?
EX
CE
IN
FP
DF
Zoom-Out Method
Are most square matrices invertible?
Are most square matrices invertible?
How often does a polynomial vanish?
Are most square matrices invertible?
How often does a polynomial vanish?
How often does a polynomial vanish?
How often does a polynomial vanish?
How often does a polynomial vanish?
With the rising sea, even distant islands disappear.
With the rising sea, even distant islands disappear.
Thank You!