Attendance at tutorials is included in the standard registration fee, so you can take them at no extra cost. All tutorials are scheduled to be 1.5 hours long and will run on Monday, June 18, 2018; the time slot (subject to change) for each tutorial is indicated below.

Diagrams 2018 has six scheduled tutorials (subject to change):

**Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning**

Presenters: Aleks Kissinger and Bob Coecke

Time: 09:00 – 10:30

Description: We provide a self-contained introduction to quantum theory using a unique diagrammatic language. Far from simple visual aids, the diagrams we use are mathematical objects in their own right, which allow us to develop from first principles a completely rigorous treatment of ‘textbook’ quantum theory. Additionally, the diagrammatic treatment eliminates the need for the typical prerequisites of a standard course on the subject, making it suitable for a multi-disciplinary audience with no prior knowledge in physics or advanced mathematics.

By subscribing to a diagrammatic treatment of quantum theory we place emphasis on quantum processes, rather than individual systems, and study how uniquely quantum features arise as processes compose and interact across time and space. We introduce the notion of a process theory, and from this develop the notions of pure and mixed quantum maps, measurements and classical data, quantum teleportation and cryptography, models of quantum computation, quantum algorithms, and quantum non-locality. The primary mode of calculation in this tutorial is diagram transformations, where simple local identities on diagrams are used to explain and derive the behaviour of many kinds of quantum processes.

This tutorial roughly follows a new textbook published by Cambridge University Press in 2017 with the same title.

**Diagrams in Kant’s Philosophy of Mathematics: Image and Intuition**

Presenter: Ofra Rechter

Time: 11:00 – 12:30

Description: Kant’s notion of an image (Bild) is at play in the examples of mathematical construction from both geometry and arithmetic that are considered in the Critique of Pure Reason. Pre-Critical texts distinguish linguistic signs from mathematical sensible means for cognition, geometrical figures and signs in numeral systems, but the Critical philosophy raises the question of what the operation signs signify. In this tutorial we articulate the

central problems about Kant’s epistemology of mathematics from the vantage point of the problematic of pure intuition as an intuition of images that instantiate mathematical concepts. Our point of departure is that a phenomenological aspect of the use made of geometrical figures in mathematical reasoning in Kant also underpins the role of numeral types in the construction of the elementary arithmetic. But, the manipulation of images in calculation or arithmetical reasoning cannot simply be a geometry of numerical signs for Kant. An attractive proposal by Michael Friedman associates the mathematical operations with what Kant calls schemata (and, thereby, via the imagination, both with time and space– the forms of sensibility, and with the intellectual synthesis under the categories). This seems to work quite neatly when the operations in question are only Euclidean postulates. Can the proposal be extended to arithmetic? to address this question we will isolate a problem that is distinctive of Kant’s philosophy of arithmetic: an inherent ambiguity in his account between successive addition as a species of synthesis and the primitive recursive binary operation of addition. By showing that the appeal to images serves Kant in bridging this ambiguity, the epistemological and the ontological accounts of a mathematical object of intuition are distinguished. Finally, time permitting, we will indicate how this result also contributes to resolving a dispute over mathematical intuition and the Kantian roots of Finitism between Charles Parsons and William Tait.

**Carroll diagrams: design and manipulation**

Presenter: Amirouche Moktefi

Time: 14:00 – 15:30

Description: The use of diagrams in logic is old. Euler and Venn schemes are among the most popular. Carroll diagrams are less known but are occasionally mentioned in recent literature. The objective of this tutorial is to expose the working of Carrol’s diagrams and their significance from a triple perspective: historical, mathematical and philosophical. The diagrams are exposed, worked out and compared to Euler-Venn diagrams. These schemes are used to solve the problem of elimination which was widely addressed by early mathematical logicians in Boole’s footsteps. Logicians worked on a general method for finding the conclusion that is to be drawn from any number of premises containing any number of terms. For this purpose, they designed symbolic, visual and mechanical devices. The significance of Venn and Carroll diagrams is better understood within this historical context. The development of logic notably created the need for complex diagrams to represent n terms, rather than merely 3. Several methods to construct diagrams for n terms, with different strategies, are discussed. Finally, the philosophical significance of Carroll diagrams is discussed in relation to the use of transfer rules. This practice is connected to recent philosophical debates on the role of diagrams in mathematical practice.

**Were “super-turing” diagrammatic reasoning competences ancient products of biological evolution?**

Presenter: Aaron Sloman

Time: 14:00 – 15:30

Description: I’ll give a highly interactive introduction to aspects of the Turing-inspired Meta-Morphogenesis project, focusing on conjectures about evolutionary processes leading to the amazing discoveries in topology and geometry by ancient mathematicians, and corresponding competences of young humans and other intelligent animals, suggesting brain processes very different from anything currently understood in AI or neuroscience. A conjectured “super-turing membrane machine” will be sketched along with requirements for further development. Whether this can be implemented as a virtual machine running on digital computing machinery is not yet clear. The tutorial will be highly interactive with many different examples discussed in detail, depending on interests of participants, e.g. examples of reasoning about affordances, using “diagrams in the mind”. More details can be found here.

**Using verbal protocols to support diagram design**

Presenter: Thora Tenbrink

Time: 16:00 – 17:30

Description: How do we know what people perceive in a diagram? A diagram can be an excellent medium for communication of complex facts and relationships. Users may be able to learn a lot just from a quick glance at a well-designed diagram. Unfortunately, what users take from a diagram may not always be the same as what its designers intended to communicate. This can have enormous consequences, ranging from misinterpretation of research outputs to false representation in the media, to the point of misguided policy decisions coming from miscommunication of central research insights.

In this tutorial, we will look at the use of verbal protocols as a tool in diagram design. The way people talk about a diagram can reveal a lot about how they understand it, what they misinterpret, and what kinds of design features could be amended to enhance clarity, ensuring successful communication.

The tutorial will start by looking at the kinds of problems that frequently arise in diagram interpretation, such as cognitive biases, misinterpretations, and effects of lack of expertise. Following a brief discussion of the value of verbal protocols in this area, we will turn to the practical aspects of verbal protocol data collection, analysis, and interpretation.

**Peirce on Diagrammatic Reasoning and Semeiotic**

Presenters: Javier Legris and Cassiano Rodrigues

Time: 16:00 – 17:30

Description: Charles Sanders Peirce (1839-1914) is one of the “grounding fathers” of mathematical logic, having developed all of the key formal results of modern logic. Starting from Boole’s algebra of logic and De Morgan’s logic of relations, Peirce developed his own system of quantifiers and relative predicates. Due to philosophical reasons, he became dissatisfied with algebraic notation for logic, developing a diagrammatic logical system of Existential Graphs. Regarding it as his masterpiece in logic, Peirce called it the logic of the future. For Peirce, all necessary reasoning is diagrammatic. Generalizing, we can say all logical inferences can be interpreted as diagrammatic experimentations upon signs. This is why Peirce thinks of logic as “semeiotic”, a general and “quasi-necessary” doctrine of signs inquiring into their active role as conveyers of meaning.

The tutorial will present Peirce’s logic and philosophy of logic, focusing more on the arguments supporting some of Peirce’s most original ideas. His distinction between logic and mathematics will be hinted at, aiming at showing why creativity and discovery have an important place in Peirce’s thought.

Specific topics: Relatives, illation, and semeiotic; quantifiers; existential graphs; theorematic and corollarial deductions; Peirce’s refusal of logicism; Notes on bibliography and current research.