On reasoning with diagrams and other analogical representations

Illustrations and common misconceptions

Apr 26, 2026

Here are some rough notes for a Beacon Unitarian humanists meeting to be held this evening (2026-04-26) on Zoom about diagrammatic reasoning — more specifically, analogical representations and analogical reasoning. I think these notes may be of interest to anyone interested in cognitive science.

We will draw heavily on the work of Aaron Sloman of the University of Birmingham, England who has written, from an AI perspective, more than anyone else on the subject. (See my encomium here.)

A prior post

Check out my 2019 post on the subject: Drawing Diagrams in the Head and with Technology: Benefits, Cognitive Mechanisms, Artificial Intelligence, Apps, and Sleep Onset Dreaming – CogZest

Terms and distinctions

We will start by distinguishing

sentences,
diagrams,
Fregean representations (roughly: representations that use application of functions to arguments to combine information items to form larger information items — recursively and with logical connectives), and
analogical representations (roughly: representations in which part of the structure of the representation maps onto the structure of that which is represented.)
world 2 (subjective) vs. world 2’ (virtual machine) vs. world 3 (objective) representations, using Karl Popper’s (essential) 3-world ontology (which I augmented Cognitive Productivity: Using Knowledge to Become Profoundly Effective)
the logicist claim: (that systems built solely on logical representations and general logical forms of inference might exhibit human-like intelligence. Counterargument here.)

Hint: the most helpful concepts in this space are Fregean and analogical reasoning. These concepts are not mutually exclusive. E.g., Fregean representations can be organized in analogical ways, e.g., a sequence of propositions matching the order in which the actions denoted by the propositions took place. A diagram can contain Fregean representations, per below. See this paper for technical definitions of these two concepts. Here we focus on analogical representations and reasoning.

Analogical representations

We’ll jump straight into some examples of analogical representations and reasoning.

The figure above shows that very abstract concepts, even the infinite, can be represented by diagrams.

The above and below show that we can reason causally with diagrams.

The next figure (taken from a prior article of mine on BSBM+ meditation which I invented) clearly illustrates many important facts about analogical representations. Parts of an analogical representation map to parts of the represented object. This also shows how a diagram can indicate sequence. It also illustrates the annotation of a diagram. (Hohol & Milkowski (2019) demonstrate the great historical importance of annotating diagrams.) It also shows that a diagram can contain sentences which are also analogical (here, the order of the sentences reflects the sequence of the procedure). What else does it show?

Can you count how many ways in which Western musical notation is analogical? The next figure might help:

Geometry and topology

The most rigorous analogical representations and forms of reasoning are in geometry and topology. If there’s time I will illustrate geometrical proofs. See also my recent article What We Cut from Education—and Why It Matters for Cognitive Science and AI (i.e., in many jurisdictions geometry has been sacrificed, and that is a shame).

Sign language and videos

Many signs are analogical. Gesture taps into an ancient part of the brain. Drawing is an extension of gesture. (Search for “gesture” in this PDF for more information about the evolution of language from a design stance.)

Conducting is a great example of analogical representations and their use (not so much about analogical reasoning, however). See this series of videos on the subject.

Videos themselves are also analogical, which is a major reason they are so helpful for learning.

Mr. Bean

Let’s jump into an example from Sloman, 2002: “Diagrams in the Mind?” Mr. Bean was on the beach, and wished to remove his underpants then to put on his swimming trunks, both without removing his trousers. How can he do this?

Here he is thinking about the problem:

Now he has gotten pretty far:

Almost done!:

Voilà!:

Watch him in action on YouTube.

Can you count how many different ways Mr. Bean could have accomplished this feat? You may be surprised by the answer. Answering this question requires extensive topological, i.e., analogical reasoning. (Fregean reasoning won’t do.) The solution can be reduced using the following diagrams which result from and support topological reasoning on the problem faced by Mr. Bean:

Questions

We will address some of the questions below. But please also bring your own questions … and diagrams (or videos).

How and why do you tend to draw diagrams for understanding something or solving problems?
What visuals do you use?
How are analogical representations used in Unitarian services (not just diagrams)?
Can you visualize the properties of infinity, such as an infinite sequence of dominoes falling over each other? How?

Why are the following claims about analogical vs. Fregean representations false?

The mind contains diagrams.
Analogical representations are continuous, Fregean representations discrete
Analogical representations are 2-dimensional, Fregean representations 1-dimensional.
Analogical representations are isomorphic with what they represent.
Fregean representations are symbolic, analogical representations non-symbolic.
Sentences in a natural language are all Fregean.
Analogical representations are complete: whatever is not represented in a picture or map is thereby represented as not existing. By contrast Fregean representations may abstract from as many or as few features of a situation as desired: if I say ‘Tom stood between Dick and Harry’, then nothing is implied about whether anyone else was there or not.
Fregean representations have a grammar, analogical representations do not.
Although digital computers can use Fregean representations, only analog computers can handle analogical representations.
Every symbolism, or representational system, must be analysed as being either analogical or Fregean.

The answers to the above are in Sloman (1975): Afterthoughts on analogical representations.

And:

Visualizing is like seeing

is countered in Sloman (2002) Diagrams in the Mind?

AI

It has been 55 years since Aaron Sloman first called for AI to pay more attention to analogical representations. There has been some progress but AI, and other cognitive sciences’, models of diagrammatic and topological reasoning are still in their infancy. Understanding these capabilities is a hard problem in cognitive science and AI.

Luc Beaudoin: CogZest

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