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Take the inductive leap

Mindful Modeler • 279 implied HN points • 30 Apr 24
🔬 Science Machine Learning Induction Prediction Philosophy
  1. In a 2-day universe, predicting the future is uncertain and relies on assumptions, highlighting the challenge of inductive reasoning.
  2. The problem of induction questions the idea that the future will always mirror the past, emphasizing the need to critically assess assumptions.
  3. Taking an inductive leap involves making predictions based on past observations and acknowledging the inherent uncertainty and need to challenge assumptions in our understanding of the world.

David Hume—Why we change our mind

Ralph Ammer • 668 implied HN points • 02 May 23
📖 Philosophy Empiricism Rationalism Induction
  1. David Hume believed that our knowledge comes from outside through observation, not from within through reason.
  2. Empiricists trust that the material world has structure through cause and effect, but Hume questioned the certainty of this trust.
  3. Deductions provide logical certainty, while inductions offer general rules based on probability.

Corridor numbers, Fibonacci numbers, and Pascal’s triangle

A Piece of the Pi: mathematics explained • 12 implied HN points • 25 Feb 24
🔬 Science Mathematics Geometry Chess Induction
  1. Corridor numbers count ways to take diagonal steps down a corridor with fixed width. The numbers in each box form Fibonacci numbers when summed vertically.
  2. Fibonacci sequence is generated by summing the previous two terms. In the context of corridor numbers, Fibonacci numbers represent different routes to specific boxes.
  3. Pascal's triangle has rows starting and ending with 1, where each entry is the sum of two nearest entries from the row above. Circular Pascal arrays relate to corridor numbers and can produce Fibonacci numbers when subtracting specific entries.

Mathematical induction

Infinitely More • 10 implied HN points • 30 Jan 24
🚌 Education Mathematics Proofs Induction
  1. Mathematical induction is a fundamental principle in mathematics, used to prove many fundamental facts in arithmetic and number theory.
  2. The common induction principle states that if a set of natural numbers contains 0 and whenever n is in the set, n+1 is also in the set, then every natural number is in the set.
  3. Strong induction allows the induction step to use multiple smaller numbers to prove a statement, and can be proven from the least-number principle.

Applications of induction

Infinitely More • 5 implied HN points • 13 Mar 24
🚌 Education Mathematics Games Induction Numbers
  1. Induction is about the impossibility of minimal counterexamples, and it comes in various forms like common induction and strong induction.
  2. Flexible use of induction is key - choose the valid form that best fits your proof.
  3. Differentiate between examples and proofs - examples can provide insight but don't prove universal statements.
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Solutions manual for NTB - 1.1 Divisibility and primality

Thái | Hacker | Kỹ sư tin tặc • 0 implied HN points • 07 Jul 09
🚌 Education Mathematics Proofs Number Theory Induction
  1. Check for divisibility between two numbers by comparing their multiples with another common multiple, often using integers.
  2. Composite integers have prime divisors that are less than or equal to the square root of the composite integer.
  3. The number of multiples of a given integer within a specified interval can be calculated using floor functions and division.