DZone
Thanks for visiting DZone today,
Edit Profile
  • Manage Email Subscriptions
  • How to Post to DZone
  • Article Submission Guidelines
Sign Out View Profile
  • Post an Article
  • Manage My Drafts
Over 2 million developers have joined DZone.
Log In / Join
Refcards Trend Reports Events Over 2 million developers have joined DZone. Join Today! Thanks for visiting DZone today,
Edit Profile Manage Email Subscriptions Moderation Admin Console How to Post to DZone Article Submission Guidelines
View Profile
Sign Out
Refcards
Trend Reports
Events
Zones
Culture and Methodologies Agile Career Development Methodologies Team Management
Data Engineering AI/ML Big Data Data Databases IoT
Software Design and Architecture Cloud Architecture Containers Integration Microservices Performance Security
Coding Frameworks Java JavaScript Languages Tools
Testing, Deployment, and Maintenance Deployment DevOps and CI/CD Maintenance Monitoring and Observability Testing, Tools, and Frameworks
Culture and Methodologies
Agile Career Development Methodologies Team Management
Data Engineering
AI/ML Big Data Data Databases IoT
Software Design and Architecture
Cloud Architecture Containers Integration Microservices Performance Security
Coding
Frameworks Java JavaScript Languages Tools
Testing, Deployment, and Maintenance
Deployment DevOps and CI/CD Maintenance Monitoring and Observability Testing, Tools, and Frameworks
Join us today at 1 PM EST: "3-Step Approach to Comprehensive Runtime Application Security"
Save your seat

The Smallest Uninteresting Number and Fuzzy Logic

John Cook user avatar by
John Cook
·
Jan. 01, 13 · Interview
Like (1)
Save
Tweet
Share
3.69K Views

Join the DZone community and get the full member experience.

Join For Free

I’ve tried to think of something interesting about the number 2013 and haven’t come up with anything. This reminds me of the interesting number paradox.

Theorem: All positive integers are interesting.

Proof: Let n be the smallest uninteresting positive integer. Then n is interesting by virtue of being the smallest such number.

The interesting number paradox is semi-serious, and so is the resolution I propose below. Both are jokes, but they touch on some serious ideas.

“Interestingness” is not an all-or-nothing property. Some numbers are more interesting than others, so perhaps we should use fuzzy logic to quantify how interesting a number is, say on a scale from 0 to 1.

For a given ε > 0, define as interesting the set of numbers whose interestingness is greater than ε. Suppose the interestingness of numbers trails off after some point. (Otherwise, if the interestingness dropped sharply, the first number after the drop would be interesting.) The largest interesting number then is barely interesting. The number one larger than a barely interesting number is even less interesting. So the proof of the interesting number paradox doesn’t apply in the continuous setting.

On a more serious note, many paradoxes in mathematics can be resolved by replacing a binary criterion with a continuous one.

For example, the sum of a trillion continuous functions is continuous, but the infinite sum of continuous functions may not be. How can that be? The problem is that we’re viewing continuity as an all-or-nothing property. If you have a series of continuous functions that converges to a discontinuous limit, the degree of continuity must be degrading. The partial sum after some large number of terms is continuous, but not very continuous. The modulus of continuity of each partial sum is finite, but is getting larger, and is infinite in the limit.

Classical statistics is filled with yes-no concepts that make more sense when replaced with continuous measures. For example, instead of asking whether an estimator is biased, it’s more practical to ask how biased it is.

Computer science is often concerned with whether something can be computed (i.e. exactly). But sometimes it’s more important to ask how well something can be computed. Many things that cannot be computed in theory can be computed well enough in practice.

All or nothing (armor)

Published at DZone with permission of John Cook, DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

Popular on DZone

  • ChatGPT: The Unexpected API Test Automation Help
  • Efficiently Computing Permissions at Scale: Our Engineering Approach
  • Tech Layoffs [Comic]
  • A Simple Union Between .NET Core and Python

Comments

Partner Resources

X

ABOUT US

  • About DZone
  • Send feedback
  • Careers
  • Sitemap

ADVERTISE

  • Advertise with DZone

CONTRIBUTE ON DZONE

  • Article Submission Guidelines
  • Become a Contributor
  • Visit the Writers' Zone

LEGAL

  • Terms of Service
  • Privacy Policy

CONTACT US

  • 600 Park Offices Drive
  • Suite 300
  • Durham, NC 27709
  • support@dzone.com
  • +1 (919) 678-0300

Let's be friends: