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Qualitative–Quantitative Reasoning: Thinking informally about formal things

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Alan Dix
http://alandix.com/papers/ICTCS-QQ-2021/
@alanjohndix @CompFoundry
Qualitative–Quantitative Reasoning
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Tiree Tech Wave
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Qualitative–Quantitative Reasoning: Thinking informally about formal things

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Keynote at ICTAC 2021: 18th International Colloquium on Theoretical Aspects of Computing, Nazarbayev University, Nur-Sultan, Kazakhstan, September 6-10, 2021.

https://alandix.com/academic/papers/ICTCS-QQ-2021/

When I first read Hardy and Wright’s Number Theory I was captivated. However, as much as the mathematics itself, one statement always stood out for me. In the very first chapter they list a number of "questions concerning primes", the first of which is whether there is a formula for the nth prime. Hardy and Wright explicitly say that this seems "unlikely" given the distribution of the series is "quite unlike what we should expect on any such hypothesis." I think most number theorists would still agree with this assertion, indeed many cryptographic techniques would collapse if it such a formulae were discovered. Yet what is this sense that the structure of primes and the structure of formulae are so different? It is not formal mathematics itself, else it would be a proof.

In engineering, computation, physics, indeed any quantitative or formal domain, the precise and provable sits alongside an informal grasp of the nature of the domain. This was certainly true in my own early and more recent work on formal modelling of human computer interaction: sometimes, as in the case of undo, one can make exact and faithful statements and proofs, but more often in order to achieve formal precision, one resorts to simplistic representations of real-life. However, despite their gap from the true phenomena, these modes, however lacking in fidelity, still give us insight.

I'm sure this will be familiar to those working in other areas where theoretical models are applied to practical problems. There is a quantum-like tension between the complexity of the world and our ability to represent it, between accuracy and precision, between fidelity and formality. Yet, we do learn about real phenomena from these simplified models, and in many contexts, from primary school estimation to scientific research we use these forms of thinking – I call this qualitative–quantitative reasoning.

This has become particularly important during Covid, when both simple formulae and massive supercomputing models offer precise predictions of the impact of specific interventions. However, even the most complex model embodies simplifications and it is when the different models lead to qualitatively similar behaviours that they are most trusted. Similar issues arise for climate change, international economics and supermarket shopping.

Qualitative–quantitative reasoning is ubiquitous, but not often discussed – almost a dirty secret for the formalist and yet what makes theory practical. There are lessons for science and for schools, challenges for visualisation and argumentation. I don’t know all of the answers, but by bringing this to the surface I know there are exciting questions.

Keynote at ICTAC 2021: 18th International Colloquium on Theoretical Aspects of Computing, Nazarbayev University, Nur-Sultan, Kazakhstan, September 6-10, 2021.

https://alandix.com/academic/papers/ICTCS-QQ-2021/

When I first read Hardy and Wright’s Number Theory I was captivated. However, as much as the mathematics itself, one statement always stood out for me. In the very first chapter they list a number of "questions concerning primes", the first of which is whether there is a formula for the nth prime. Hardy and Wright explicitly say that this seems "unlikely" given the distribution of the series is "quite unlike what we should expect on any such hypothesis." I think most number theorists would still agree with this assertion, indeed many cryptographic techniques would collapse if it such a formulae were discovered. Yet what is this sense that the structure of primes and the structure of formulae are so different? It is not formal mathematics itself, else it would be a proof.

In engineering, computation, physics, indeed any quantitative or formal domain, the precise and provable sits alongside an informal grasp of the nature of the domain. This was certainly true in my own early and more recent work on formal modelling of human computer interaction: sometimes, as in the case of undo, one can make exact and faithful statements and proofs, but more often in order to achieve formal precision, one resorts to simplistic representations of real-life. However, despite their gap from the true phenomena, these modes, however lacking in fidelity, still give us insight.

I'm sure this will be familiar to those working in other areas where theoretical models are applied to practical problems. There is a quantum-like tension between the complexity of the world and our ability to represent it, between accuracy and precision, between fidelity and formality. Yet, we do learn about real phenomena from these simplified models, and in many contexts, from primary school estimation to scientific research we use these forms of thinking – I call this qualitative–quantitative reasoning.

This has become particularly important during Covid, when both simple formulae and massive supercomputing models offer precise predictions of the impact of specific interventions. However, even the most complex model embodies simplifications and it is when the different models lead to qualitatively similar behaviours that they are most trusted. Similar issues arise for climate change, international economics and supermarket shopping.

Qualitative–quantitative reasoning is ubiquitous, but not often discussed – almost a dirty secret for the formalist and yet what makes theory practical. There are lessons for science and for schools, challenges for visualisation and argumentation. I don’t know all of the answers, but by bringing this to the surface I know there are exciting questions.

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Qualitative–Quantitative Reasoning: Thinking informally about formal things

  1. 1. Alan Dix http://alandix.com/papers/ICTCS-QQ-2021/ @alanjohndix @CompFoundry Qualitative–Quantitative Reasoning thinking informally about formal things
  2. 2. Tiree Tiree Tech Wave https://tireetechwave.org/
  3. 3. the foundry building mission community
  4. 4. today I am not talking about … • physicality and product design • intelligent internet interfaces • long tail of small data • deep digitality • digital thinking • now • digital light • digital humanities • virtual crackers and slow time • modeling dreams, regret and the emergence of self
  5. 5. Number theory and things …
  6. 6. arithmetic mathematics ?
  7. 7. the qualitative understanding of quantitative phenomena
  8. 8. Formal models of interactive systems
  9. 9. Oxymoron? 𝛌 ? = https://en.wikipedia.org/wiki/File:The_Adicts_2011_SO36_03.jpg
  10. 10. PIE model – abstract properties predict e.g. predictability:
  11. 11. undo
  12. 12. more complex undo – the cube
  13. 13. PIE model – abstract properties predict e.g. predictability:
  14. 14. Insight! Formal model often not complete or accurate … but still gives insight predict
  15. 15. Fully Charged
  16. 16. Electrostatically-charged crop sprays
  17. 17. Above canopy model ground at 0 V open boundary condition complex sprayer at high voltage + 1000s V charged spray drops model path, speed and space voltage due to drop cloud ?
  18. 18. Within Canopy Model crop modelled as straight sides ground at bottom +V – charged spray enters crop canopy from above; speed and charge determined by above canopy model.
  19. 19. Within Canopy Model crop modelled as straight sides ground at bottom +V – charged spray enters crop canopy from above; speed and charge determined by above canopy model.
  20. 20. Misses crop (Class I)
  21. 21. Good spread (Class II)
  22. 22. All at the top (Class III)
  23. 23. Connect and re-parameterise + 1000s V above crop model within crop model dimesionless parameters
  24. 24. Design/engineering interventions X
  25. 25. Order of magnitude day to day Ocean and drops Covid and climate change
  26. 26. an ocean is made of drops Covid – masks, etc. – action action minimal impact – together reduce growth climate & pollution – each car ride, plastic cup, … minimal difference – together, each year … 1.2 trillion tonnes of ice melt 8 million tonnes of ocean plastic
  27. 27. Order of magnitude formal Algorithmic complexity Linear programming and sorting What is computation?
  28. 28. what is computation? information theory entropy independent of mechanism algorithmic complexity measures are system dependent sorting? – n log n lower bound information proof – works for oracles Galois theory feels connected?
  29. 29. standard Galois field extension ℚ ℝ ℚ(√2) computational Galois sets ℚ0(√2) ℚ∞(√2) ℚ2(√2) ℚ1(√2) ℚ3(√2) … = = reals rationals rationals extended with solution of x2=2 three √2 allowed plain rationals two √2 allowed one √2 allowed any number of √2 allowed
  30. 30. Numerical models As well as crop spraying … Covid serial interval
  31. 31. Viral metrics R – how many? Serial interval – how long? time
  32. 32. Viral metrics R – how many? Serial interval – how long? not plain disease metrics … • about disease in a particular setting: housing, social culture, precautions • interlinked time
  33. 33. Serial interval – 3 measures Forward (standard) – How long to next infection Backward – How long since infected Effective – During exponential growth R t SI
  34. 34. Infection time not uniform ⇒ during growth (R > 1) backward SI < effective SI < forward SI
  35. 35. Monotonic reasoning Shops to politics
  36. 36. Monotonic arguments Change in shop A – b < A 2D – Poincare property
  37. 37. Factory Automation
  38. 38. ↑ more automation ↓ less labour needed ↑ higher productivity ↑ higher prosperity ↓ less employment ↓ lower wages ? people better off ↑ ↓
  39. 39. ↑ more automation ↓ decline of other companies ↑ more competative ↑ growth of company ↓ less jobs elsewhere ↑ more jobs there ? overall employment ↑ ↓
  40. 40. Formalising QQR Fuzzy logic, Bayesian methods Allen’s interval algebra
  41. 41. A B A B precedes meets A B A B starts finishes A B A B contains overlaps A B equal Allen's interval algebra
  42. 42. Call to arms
  43. 43. climate change Brexit QQR critical for understanding … in short …. being a C21 citizen covid
  44. 44. Alan Dix http://alandix.com/QQR/ @alanjohndix @CompFoundry Qualitative–Quantitative Reasoning thinking informally about formal things

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