Summary

• Two algorithms
• Math and learning from data 
• Implications

This talk is partly an adaptation of the YouTube series "Learning to See" by Welch Labs.

Counting fingers

Imagine we want to teach a machine how to count fingers.

After all, counting is the foundation of arithemetic. How might we do this?

Knowledge Engineering

We could just define what a finger looks like:

Memorization

Fine. We can't define what a finger looks like, so let's take the training examples to be rules:

Memorization

On the training data, it performs well!

Knowledge engineering: 10% 

Memorization: 100%

Memorization

On the test set, not so well:

Knowledge engineering: 14% 

Memorization: 11%

Memorization with 10x the examples: 21%

Incoming Feynman quote...

To learn is to generalize

Lessons from Machine Learning

To learn is to generalize.

To learn from data, we have to make assumptions on the data.

What is the data?

Math as a learning process

Intuition

"data"

Formalism

"generalizations"

• physical world
• conjectures
• open problems
• questions
• math itself

• theorems
• axioms
• definitions
• frameworks

Proof

Abuse of notation

Child

Mathematician

Summary of the talk.

Math as a learning process

To learn from data, we have to make assumptions on the data.

Human intuition as foundations

• Explains why: ​​
  • Why we might prefer some axiomatic systems over others
    • Why the axioms must be true
  • Why the child and mathematician are different: learning
    • Engineers don't need to know abstract math
    • Children intuitively know what numbers are.
  • Generalizations of the same math can lead to different results
    • Different notations and interpretations
• Problems:
  • Non-objective, different between mathematicians
  • Not rigorous.
    • But the formalization of intuition is rigourous! This is what math is all about!

Conclusion

• Mathematics is about generalizing things in a rigorous way.
  • Really a learning process
  • Generalizing from data (human intuition)
  • The physical world is the best axiom
• A good argument for structuralism.