Updated rough draft systems | complexity | cybernetics reading list

See my post on LinkedIn (replicated below) and join the discussion there:
https://www.linkedin.com/posts/antlerboy_rough-draft-systemscomplexitycybernetics-activity-7246779585235664896-64Xz

pdf: https://www.dropbox.com/scl/fi/85zlt0t6ph8qarx7d7gic/2024-09-27-rough-draft-systems-thinking-reading-list-v1.1BT.pdf?rlkey=3rfavacsy4n6sl8j0pyedph1q&st=qagh1418&dl=0
Commentable Google Doc: https://docs.google.com/document/d/1Tt8GgQQj4Qw4HnR7DxKeF370o_HlDlpv/edit?usp=sharing&ouid=115526108239573817578&rtpof=true&sd=true

How do you get into systems | complexity | cybernetics?

Here’s my rough reading list.

There are a lot of answers to the question, many of them connecting with some kind of disjointing break from ‘normal’ ways of seeing and being. Anything from being bullied at school to being dyslexic. Being in an outsider group. Naively applying thinking from one domain to another. Studying a technical problem long enough to suddenly see it in a completely different light – then either have your breakthrough celebrated or rejected.

It isn’t some mystic thing and it doesn’t require to you break from polite society. But it is one of the richest, weirdest, most diverse and challenging, inspiring and confounding, confronting and validating things you can study.

I’m often asked for a reading list for people interested in the field, and I usually suck my teeth. Some of the books are engaging, insightful, humorous, relevant. Others are dry as old twigs but less likely to kindle a spark.

Really, it depends on you and your context – as David Ing says, it’s better to talk of the thinkers and their individual constellations of interests, history, learning, and personal tendencies than it is to talk of schools and fields and separate places.

And even presenting this reading list, I’d say that I’d recommend Terry Pratchett, Douglas Adams, Ursula K Le Guin, Italo Calvino, Jorge Luis Borges, Star Trek, old 20th Century Sci-Fi and Apartheid-era South African writing, art movies and music more – if you happen to be a bit like me. You’ll find your thing, if you’re interested.

But. The books are there – and many of them are *really good*. Top ones I’d recommend came out this decade

  • Hoverstadt’s Grammar of Systems
  • Jackson’s Critical Systems Thinking: A practitioner’s Guide
  • Opening the box – a slim little thing from SCiO colleagues
  • Essential Balances by Velitchkov

The attached list is a bit systems-practice focused. It is also too long and incomplete and partial simply for lack of time and energy.

There are *so many* flavours of systems thinking / complexity / cybernetics – do yourself a favour and don’t flog through stuff that doesn’t work for you, find things that bring your mind alive. Start with the articles and skim through.

But do start, because you will find in here the thinking and tools to find better ways of doing things for organisations, societies, the ecosystem, for people – and a lot of fun.

Tip: to save the pdf, hover over the image of the first page and find the rectangle bottom right – click that and it should go full screen. Top right you’ll have a download option, which when clicked will then resolve into a download button… (which might then open in your browser, but at least as a proper pdf you can save).

So… deep breath… what would you recommend? What do you think is missing?

#systems-thinking

How I introduce stock and flow models – Christian Moore Anderson (2024)

How I introduce stock and flow models – Christian Moore Anderson

Recursivity and Contingency- Yuk Hui (2021

From Facebook, Örsan Şenalp  in the group The Ecology of Systems Thinking

https://www.facebook.com/groups/ecologyofsystemsthinking/?multi_permalinks=26150001584652305&notif_id=1773791715607197&notif_t=group_activity&ref=notif

Title: Recursivity and Contingency
Authors: Yuk Hui
Publisher: Rowman & Littlefield
Series: Media Philosophy
Year: 2019

Memory of the World Library
https://library.memoryoftheworld.org/?fbclid=IwY2xjawQncPZleHRuA2FlbQIxMQBzcnRjBmFwcF9pZBAyMjIwMzkxNzg4MjAwODkyAAEe8zF21smBi5vXWKnUmxKWA84OfXK3wY8i2uZOlM4JyVuFmPdnRgvSIDdVSmw_aem_RDnFs0COGbHZMBKbYYm_YQ#/book/0519e7e7-b939-492e-aff0-a3ca0ac4bc98

A response to The Inner Work of System Leadership – Kania, Ruparell, Senge, and Hamilton (2026)

On LinkedIn, John Kania posted a followup to The Dawn of Systems Leadership (https://ssir.org/articles/entry/the_dawn_of_system_leadership – can it be I did not link that here):

https://www.linkedin.com/posts/john-kania-1a294020_the-inner-work-of-system-leadership-activity-7437517239865606144-7i91?utm_source=share&utm_medium=member_desktop&rcm=ACoAAACuq-oBecVFDW6PCf3lkoG-peMeuLBeoho

In a newly published piece by the Garrison Institute called The Inner Work of System Leadership my co-authors Peter Senge, Radha Ruparell and Hal Hamilton and I explore in depth the connection between inner work and outer change.

Coming 10 years after Peter, Hal and I published the popular leadership piece, The Dawn of System Leadership in Stanford Social Innovation Review, our new article elevates inner work for system leadership and highlights how the inner work of trauma healing is essential for system leaders but often goes unattended to for a myriad of reasons.

Engaging in trauma healing work helps leaders develop more capacity to see, acknowledge, and address historical, intergenerational, and collective trauma that resides in systems and impacts others. Without doing this work, system leaders will find transformational systems change to be elusive, if not impossible.

I invite you to read the new article and I welcome your responses. How important do you think it is for leaders to do their inner work and trauma healing work?


The new paper:


My response:

Also at https://www.linkedin.com/posts/antlerboy_the-inner-work-of-system-leadership-activity-7439947524905328640-YjCc (full LinkedIn post)

And in comments to the original:

https://www.linkedin.com/feed/update/urn:li:activity:7437517239865606144?commentUrn=urn%3Ali%3Acomment%3A%28activity%3A7437517239865606144%2C7439946998163795970%29&dashCommentUrn=urn%3Ali%3Afsd_comment%3A%287439946998163795970%2Curn%3Ali%3Aactivity%3A7437517239865606144%29

It’s a good piece and will deserve its popularity – but I think it lacks the real grit in the oyster.

Your question begs the question, doesn’t it?

“How important do you think it is for leaders to do their inner work and trauma healing work?” well, yes, put it that way – obviously, VERY important. It reminds me of a phrase I loved from the wave of sensemaking, metarational, new thinking podcasts just before and into COVID: ‘we’ll never create a world of holistic wellbeing from our broken states.

And I’m not against it. Of course it matters. No one wants leaders who are blind to themselves or others.

But that’s not quite the same as the stronger claim being smuggled in: that transforming systems begins with transforming ourselves.

And let me contrast two alternative statements:

  • ‘deeply flawed, traumatised, broken, sometimes immoral people can achieve great good in the world – indeed, great good has never been done any other way’
  • ‘it’s not doing the work on yourself that changes the world, it’s doing the work in the world that changes you’

Similar rhetorical devices, I fancy similar rhetorical power, but rather contrary meaning. I am crafting those sentences only for effect, but/and both feel closer to reality than a prefigurative ideal where better selves produce better systems.

I think the paper is blending very good and strong arguments and slipping in some untested and unchallenged assumptions.

A defensible claim:

‘The way we show up shapes the systems we are trying to change’

quietly becomes:

‘Inner work is essential for transformational systems change’

That leap isn’t evidenced, it’s asserted.

And it matters, because it risks narrowing the field of vision. You end up with a picture of systems change that centres on a fairly specific set of behaviours: openness, empathy, reflection, healing. All good things. But also a very WEIRD, middle-class slice of human possibility.

What drops out is power. Conflict. Dominance. Incentives. Enforcement. Violence. Anger. The fact that systems often work precisely because they constrain behaviour, not because people are aligned or healed.

There’s a long-standing tendency in systems thinking to slide into ‘if we could all just relate better, things would improve’. Sometimes true, especially in the room. But as a theory of change for large-scale systems, it’s thin.

Or more bluntly, as they used to say (about voting): if systems change could change things, it would be illegal.

That’s the gap I think this work is circling. A real and important insight – that inner state and outer system aren’t separate – alongside an underdeveloped account of how change actually happens when interests collide and power is at stake.

If we don’t face that directly, the space gets filled by something else – and, I observe, often a kind of bourgeois conspirituality: psychologically literate, ethically serious, but politically and operationally evasive. Spiritual bypassing, which is a form of complicity.

Inner work matters. But it’s not a substitute for a theory of power, or a method for acting in the world when things get rough. And without that, systems change stays safe, and therefore limited.

I encourage everyone to read the piece – and read it really critically, and to listen as a companion to the two parts of this podcast:

https://www.conspirituality.net/episodes/brief-graeber-bannon-anarchism-leninism/


Relevant links:

SCiO Finland – Understanding and Achieving Systemic Governance: Lessons from Finland – 10:30 EET (Finnish time), March 26 2026 – with Ray Ison and Tim Logan, moderated by Janne Korhonen

Thu, Mar 26th, 2026
10:30 – 11:45  EET
WORKSHOP
SCiO Finland
All welcome

Date & Time
Thu, Mar 26th, 2026 – 10:30 – 11:45  EET

Location
This is an online event


Pricing Info
Free

Languages spoken
English

Access
All welcome

We are at a critical juncture where traditional, hierarchical governance systems often struggle to navigate the complexities of the climate emergency and mounting social inequalities.

You are cordially invited to a special webinar launch event hosted by SCiO Finland to explore the findings of a recent systemic co-inquiry: “Understanding and Achieving Systemic Governance: Lessons from Finland”.

Finland is internationally recognised for its innovative “humble governance” and trust-based autonomy, yet these systemic approaches face significant challenges in becoming “sticky” within traditional bureaucratic structures. We will dive into:

Core Principles: Moving from government as a controller to an enabler.
Scaling Learning: Why we should focus on scaling capability rather than just fixed solutions.
The “Systems Mafia”: The vital role of informal networks in sustaining systemic change.
Sustainability: How to institutionalise systemic ideas so they survive political cycles and economic pressures.
In this session, authors Ray Ison (Professor Emeritus, The Open University) and Tim Logan (Partner, Good Impact Labs) will outline the key ideas from their research conducted in Finland between 2024 and 2025. The discussion will be moderated by Janne J. Korhonen.

Understanding and Achieving Systemic Governance: Lessons from Finland | SCiO – Systems and Complexity in Organisation
https://www.systemspractice.org/events/understanding-and-achieving-systemic-governance-lessons-finland

Ray Ison – Transforming human activity systems: towards an agriculture for improving human-biosphere relations – tomorrow, 17 March 2026, 10am (Swedish time, I’m assuming)

Welcome to this webinar with Professor Emeritus Ray Ison, talking about how we can transform human activity systems towards a more improved human-biosphere agriculture. The webinar is hosted by SLU Global and ELLS . Please register below for zoom-link.

Date: 17 March 2026

Time: 10:00 – 11:15

Language: English

Organiser: SLU Global

Co-organiser: ELLS

Location: Online

Transforming human activity systems: towards an agriculture for improving human-biosphere relations | slu.se
https://www.slu.se/en/calendar/2026/03/ray-ison/

Cameron Tonkinwise – Service Design & Social Complexity – The Complexity Lounge – Tuesday March 17, 7pm UK time


Tuesday, Mar 17 · 7:00 PM to 9:00 PM GMT



Online event

Needs a link

Cameron Tonkinwise – Service Design & Social Complexity
Hosted by Jocko S.

Cameron Tonkinwise – Service Design & Social Complexity, Tue, Mar 17, 2026, 3:00 PM | Meetup
https://www.meetup.com/complexitylounge/events/313669353/?eventOrigin=group_upcoming_events

Differential Logic • The Logic of Change and Difference

Differential logic is the logic of variation — the logic of change and difference.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple case of a differential logical calculus is furnished by a differential propositional calculus, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

cc: Academia.eduCyberneticsLaws of FormMathstodon
cc: Research GateStructural ModelingSystems ScienceSyscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

A VSM governance tool and agricultural paradigm change for climate and ecology – Jenkinson (2026) (paywalled)

TECHNICAL PAPER| MARCH 06 2026
A VSM governance tool and agricultural paradigm change for climate and ecology
Angus Jenkinson

Author & Article Information
Kybernetes 1–25.
https://doi.org/10.1108/K-06-2025-1586
Article history
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Purpose
This study offers transformation pathways through the intersections of enterprise governance and farming in response to climate and ecological crises (COP21, COP28 et al.). The analysis further proposes Virtuoso as a novel integration capability for the Viable System Model (“VSM”), Stafford Beer’s systemic methodology for enterprise management (Beer, 1985; Jenkinson, 2022). It thereby contrasts farming models and their ways of seeing.

A VSM governance tool and agricultural paradigm change for climate and ecology | Kybernetes | Emerald Publishing

https://www.emerald.com/k/article-abstract/doi/10.1108/K-06-2025-1586/1344986/A-VSM-governance-tool-and-agricultural-paradigm?redirectedFrom=fulltext

On LinkedIn https://www.linkedin.com/posts/angusjenkinson_virtuoso-biodynamicfarming-fertilizers-activity-7437173834631442432-Hmq4?utm_source=share&utm_medium=member_desktop&rcm=ACoAAACuq-oBecVFDW6PCf3lkoG-peMeuLBeoho, Angus says:


Angus Jenkinson

Very happy to share this paper I just published about different farming models and their impact on nature and climate (A VSM governance tool and agricultural paradigm change for climate and ecology). it shows how this relates to ways of seeing the world and managing a company or enterprise. This features the VSM, the viable system model developed by Stafford Beer, whose centennial is this year. It takes a deep look at identity as the guiding star and model.

But what really excites me is its illustration of the immense potential the right kind of farming has. The key to the life and economic health of the farm is soil management and that depends on understanding how to cherish it so that it will support farmer, food, and future.

You see the future is very bleak if farming does not get its act fully together, so I am delighted to showcase genuine best practice in the case study. The featured farm is sequestering around ten times more carbon than it emits

https://lnkd.in/e5xsMMNq

Findings

Farming has radically altered the planetary biosphere (Ellis et al., 2010). The IC model is a major contributor to global climate and biodiversity challenges through its degenerative cycle of soil degradation, input dependency, and declining resilience (Montgomery, 2017, 39–40, 80–81). BD demonstrates that economically sound, mitigation and regeneration is possible, exceeding UN COP21 soil organic carbon (“SOC”) targets, a key indicator of climate change (Gantlett, 2021, 2022, 2024, 2025). Virtuoso articulates their contrasting identities and operational logics, revealing latent potential for agriculture and the VSM.

Launch Event: What does it take to sustain place-based change? Renaisi-TSIP, 21 April 2026, London UK


Follow
Coin Street Neighbourhood CentreLondon, England
Tuesday, Apr 21 from 11 am to 3 pm

Launch Event: What does it take to sustain place-based change? Tickets, Tuesday, Apr 21 from 11 am to 3 pm | Eventbrite
https://www.eventbrite.com/e/launch-event-what-does-it-take-to-sustain-place-based-change-tickets-1982189954513?aff=oddtdtcreator

Northeast Regional Conference on Complex Systems 2026 – on now!

ByNERCCS

University of RochesterRochester, NY

Wednesday, Mar 11 at 12 pm to Friday, Mar 13 at 12:30 pm

Weirdly, they stopped selling $15 onine tickets but program

And follow on twitter:

https://twitter.com/hashtag/NERCCS2026?src=hashtag_click&f=live

Cybernetic Praxis, Part II: Toward a Reconstruction of Construction – Wood (2026)

Ben Wood, PMP

Project Consultant | Builder | Systems Thinker

Cybernetic Praxis, Part II: Toward a Reconstruction of Construction

Ben Wood, PMP
Project Consultant | Builder | Systems Thinker

(1) Cybernetic Praxis, Part II: Toward a Reconstruction of Construction | LinkedIn
https://www.linkedin.com/pulse/cybernetic-praxis-part-ii-toward-reconstruction-ben-wood-pmp-xr2we/?trackingId=0opg5Q4iSPSmttxWTBe%2FYg%3D%3D

How do we find the “why”? Vornov (2026)

How do we find the “why”? – On Deciding . . . Better 3.0

Mindwalk – a film by Bernt Capra

Recommended as ‘the only film about systems thinking’ on LinkedIn…

MINDWALK – YouTube

ChatGPT says:

Mindwalk (1990) is an unusual film. It is almost entirely a long philosophical conversation rather than a conventional story. The director is Bernt Amadeus Capra, and the ideas come largely from his brother, physicist Fritjof Capra, especially the book The Turning Point (1982). (Wikipedia)

The film runs about 112 minutes and stars Liv Ullmann, Sam Waterston, and John Heard. It was filmed largely on location at Mont-Saint-Michel in Normandy, whose medieval architecture and tides serve as a kind of symbolic backdrop to the discussion. (Wikipedia)

The basic premise
Three people wander around the island and talk:

Sonia Hoffman – a Norwegian physicist who left her research career after realising it was feeding weapons development.
Jack Edwards – an American politician recovering from a failed presidential campaign.
Thomas Harriman – a poet and former political speechwriter who is disillusioned with politics. (Wikipedia)

The entire film is essentially their walk and conversation. They move through cloisters, terraces, and courtyards while arguing about science, politics, ecology, economics, and culture.

The core idea
The film’s intellectual centre is systems thinking. Sonia argues that modern society still thinks in a mechanistic, reductionist way inherited from Newtonian science. Instead, she proposes a holistic or systems view, where ecological, social, technological, and economic processes are interconnected networks. (What’s After the Movie?)

So the conversation ranges across things like:

• quantum physics and uncertainty
• ecological limits and environmental damage
• how political decisions ripple through complex systems
• the relationship between science, ethics, and power

Rather than giving solutions, the film tries to shift the frame of thinking—from isolated problems to interacting systems. (What’s After the Movie?)

Style and structure
It’s basically a “conversation film”. The drama is intellectual rather than plot-driven. The closest comparisons are films like My Dinner with Andre or, later, Linklater’s dialogue films. Critics often note that the risk with such films is that characters become mouthpieces for ideas. (sensesofcinema.com)

To counter that, the film uses:

• the striking setting of Mont-Saint-Michel
• shifting camera compositions of the three characters
• poetry (including a reading of Neruda at the end)

These elements keep the film from becoming purely didactic.

Reception
Critics often say the same thing: it’s thoughtful but demanding. One review described it as an engaging but dialogue-heavy film that mostly appeals to viewers already sympathetic to its ecological and systems-thinking ideas. (Wikipedia)

Why it still gets talked about
Despite being small and somewhat obscure, Mindwalk became a cult favourite in intellectual and ecological circles because it tried to translate systems theory and modern physics into a human conversation about society.

For people interested in systems thinking, complexity, ecology, or the philosophy of science, it’s almost a cinematic essay rather than a typical film.

If you want, I can also explain the key systems ideas in the film (Capra’s worldview, deep ecology, network thinking, etc.). That’s really the heart of it.

Differential Logic • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition f = pq : X \to \mathbb{B} in the following way.

The next venn diagram shows the differential proposition \mathrm{d}f = \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B} we get by extracting the linear approximation to the difference map \mathrm{D}f = \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B} at each cell or point of the universe X.  What results is the logical analogue of what would ordinarily be called the differential of pq but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for \mathrm{d}f whenever it’s necessary to single it out.

Tangent Map d(pq) : EX → B
\text{Tangent Map}~ \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

\begin{array}{rcccccc}  \mathrm{d}(pq)  & = &  p & \cdot & q & \cdot &  \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)}  \\[4pt]  & + &  p & \cdot & \texttt{(} q \texttt{)} & \cdot &  \mathrm{d}q  \\[4pt]  & + &  \texttt{(} p \texttt{)} & \cdot & q & \cdot &  \mathrm{d}p  \\[4pt]  & + &  \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0  \end{array}

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

\begin{matrix}  \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)}  & = &  \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)}  & + &  \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~}  \\[4pt]  dp  & = &  \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~}  & + &  \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)}  \\[4pt]  \mathrm{d}q  & = &  \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~}  & + &  \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~}  \end{matrix}

Capping the analysis of the proposition pq in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map \mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B}, which happens to be linear in pairs of variables.

Remainder r(pq) : EX → B
\text{Remainder}~ \mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B}

Reading the arrows off the map produces the following data.

\begin{array}{rcccccc}  \mathrm{r}(pq)  & = & p & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q  \\[4pt]  & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q  \end{array}

In short, \mathrm{r}(pq) is a constant field, having the value \mathrm{d}p~\mathrm{d}q at each cell.

Resources

cc: Academia.eduCyberneticsLaws of Form • Mathstodon (1) (2)
cc: Research GateStructural ModelingSystems ScienceSyscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

Differential Logic • 17

Enlargement and Difference Maps

Continuing with the example pq : X \to \mathbb{B}, the following venn diagram shows the enlargement or shift map \mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B} in the same style of field picture we drew for the tacit extension \boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}.

Enlargement E(pq) : EX → B
\text{Enlargement}~ \mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}

\begin{array}{rcccccc}  \mathrm{E}(pq)   & = & p & \cdot & q & \cdot &  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}  \\[4pt]  & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot &  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~}  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot &  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)}  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot &  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~}  \end{array}

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields \boldsymbol\varepsilon f and \mathrm{E}f, both of the type \mathrm{E}X \to \mathbb{B}, is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension \boldsymbol\varepsilon f and the enlargement \mathrm{E}f are in a sense dual to each other.  The tacit extension \boldsymbol\varepsilon f indicates all the arrows out of the region where f is true and the enlargement \mathrm{E}f indicates all the arrows into the region where f is true.  The only arc they have in common is the no‑change loop \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} at pq.  If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of \mathrm{D}(pq) = \boldsymbol\varepsilon(pq) + \mathrm{E}(pq) shown in the following venn diagram.

Differential D(pq) : EX → B
\text{Difference}~ \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}

\begin{array}{rcccccc}  \mathrm{D}(pq)   & = & p & \cdot & q & \cdot &  \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}  \\[4pt]  & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot &  \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~}  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot &  \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~}  \\[4pt]  & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot &  \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~}  \end{array}

Resources

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