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Irina Starikova

UFMG - The Federal University of Minas Gerais, Philosophy, Post-Doc

Slovak Academy of Sciences, Department of Analytic Philosophy, Visiting Researcher

University of Sao Paulo, Dept Of Philosophy, Post-Doc

National Research University “Higher School of Economics” (HSE), Moscow, Russia, The International Laboratory for Logic, Linguistics and Formal Philosophy (LLFP), Faculty Member

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Irina Starikova worked as a senior researcher at the The International Laboratory for Logic, Linguistics and Formal Philosophy (LLFP), High School of Economics (HSE), Moscow. She completed her postdoctoral research at the University of São
Paulo and the University of Minas Gerais. Her research areas are the philosophy of mathematical practice,
epistemic and cognitive aspects of geometrical thinking, epistemic roles of
visualisations in mathematics, thought experiments and mathematical
aesthetics. Her work focuses on examples from current advanced mathematics
such as Gromov's geometric approach to group theor. She studied mathematics at Novosibirsk Phys-Maths College and the
University at Akademgorodok Scientific Center and completed her PhD in
philosophy of mathematics at the University of Bristol, UK.
Supervisors: Vitaly Tselischev, Hannes Leitgeb, Osvaldo Pessoa Jr., Leon Horsten, and Abilio Rodriguez
Phone: +447944219088

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Johannes Zachhuber

University of Oxford

Noel B. Salazar

KU Leuven

University of California, Santa Barbara

Harvard University

Universidad Nacional de Colombia (National University of Colombia)

University of Genova

Oleg Yu Vorobyev

Siberian Federal University

Eszter Hargittai

University of Zurich, Switzerland

Simon O'Sullivan

Goldsmiths, University of London

University of Cologne

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Papers by Irina Starikova

Thought Experiments in Mathematics: From Fiction to Facts

Springer eBooks, Dec 31, 2022

Like science and philosophy, thought experiments in mathematics link a problem to new epistemic r... more Like science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that were unavailable earlier. Performing an imaginary scenario can involve counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with proofs, diagrams and calculative procedures. Reflection about thought experiments helps to realize limits and opportunities of mathematical thinking. The analysis of examples suggest a broader understanding of &quot;mathematical&quot;, &quot;thought&quot; and &quot;experiment.&quot;

Thought experiments in mathematics: from fiction to facts

Like science and philosophy, thought experiments in mathematics link a problem to new epistemic r... more Like science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that were unavailable earlier. Performing an imaginary scenario can involve counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with proofs, diagrams and calculative procedures. Reflection about thought experiments helps to realize limits and opportunities of mathematical thinking. The analysis of examples suggest a broader understanding of "mathematical", "thought" and "experiment."

Policies for including disabled students in Russia

Growing attention to the issue of higher education for disabled people in Russia reflects the urg... more Growing attention to the issue of higher education for disabled people in Russia reflects the urgent and problematic character of the situation. This paper, based on current research in the field, proposes policy reforms on a regional level elaborated in the framework of the Higher Education Working Group of the International Policy Fellowship programme. Proposals include: 1. creating a service or appointing an employee responsible for the assistance of disabled students; 2. making the necessary architectural changes to ensure the accessibility of university buildings; 3. continuity of education for disabled children; 4. active efforts by universities towards communicating with potential students; 5. creating informational basis including information on rights and other relative issues for disabled persons in the sphere of higher education; 6. campaign for community awareness; 7. teachers' training on working with disabled students.

Revisiting the mutilated chessboard or the many roles of a picture

by Jean Paul Van Bendegem and Irina Starikova

The impact of connectivity on the memory capacity and the retrieval dynamics of Hopfield-type networks

Most models of neural associative memory have used networks with broad connectivity. However, thi... more Most models of neural associative memory have used networks with broad connectivity. However, this seems unrealistic from a neuroanatomical perspective. A simple model of associative memory with emergent properties was introduced by Hopfield (1982). We choose this widely known model to investigate the impact of connectivity on the storage capacity and the retrieval dynamics in Hopfield-type networks. We used sparse topologies where only a small fraction of possible connections were actually active. For these diluted topologies, we tested different kinds of architecture; namely, randomly connected networks, regularly connected networks, small-world networks, and modular networks. Computer experiments reveal that, among the diluted topologies, the modular architectures that exhibit a relatively high clustering coefficient and whose average characteristic path lengths and total physical wiring lengths are short, produce much better results in terms of pattern storage and generalization...

Thought Experiments in Mathematics

The Routledge Companion to Thought Experiments, 2017

It is not news that we often make discoveries or find reasons for a mathematical proposition by t... more It is not news that we often make discoveries or find reasons for a mathematical proposition by thinking alone. But does any of this thinking count as conducting a thought experiment? The answer to that question is "yes", but without refinement the question is uninteresting. Suppose you want to know whether the equation [ 8x + 12y = 6 ] has a solution in the integers. You might mentally substitute some integer values for the variables and calculate. In that case you would be mentally trying something out, experimenting with particular integer values, in order to test the hypothesis that the equation has no solution in the integers. Not getting a solution first time, you might repeat the thought experiment with different integer inputs. The fact that there are such mundane thought experiments is no surprise and does not answer the question we are really interested in. 2 The numerical thought experiment just given involves nothing more than applying mathematically prescribed rules (such as rules of substitution and calculation) to selected inputs. It would be more interesting if there were mathematical thought experiments in which the experimental thinking goes beyond application of mathematically prescribed rules, by using sensory imagination as a way of eliciting the benefits of past perceptual experience. 3 In what follows we will try to show that there are such thought experiments and to assess their epistemic worth. Our method will be to present some candidate thought experiments with what we hope is enough background explanation and in sufficient detail for you, the reader, to perform the relevant mental operations yourself; without this participation the paper will be neither convincing nor engaging. We have tried to avoid run of the mill 1 I would like to thank the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) and the Russian Foundation of Basic Research (RFBR). 2 For this reason we find that the category of thought experiments as characterised by Jean-Paul Van Bendegem in "Thought experiments in mathematics: anything but proof" Philosophica 72, (2003), pp. 9-33 to be too broad. 3 For a different focus, see Eduard Glas, "On the role of thought experiments in mathematical discovery" in J. Meheus and T. Nickles (eds.), Models of Discovery and Creativity, (Springer 2009). Glas says that "imagery, mental or experiential, is not essential" to the aspect of thinking that he counts as thought experiment (even when accompanied by imagery). For this reason, the kinds of thinking that we discuss in this paper do not fall under what Glas counts as thought experiment.

Picture-proofs and platonism

Croatian Journal of Philosophy, 2007

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kant... more This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the "seeing" of mathematical ideas by means of pictures, diagrams, thought experiments, etc. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss -James Robert Brown's recent book Philosophy of Mathematics, in particular, his primarily examples and analogies. I then consider some alternatives and counterarguments, namely John Norton's opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification,...

From Practice to New Concepts: Geometric Properties of Groups

Philosophia Scientiae, 2012

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant ... more Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation

Aesthetic Preferences in Mathematics: A Case Study†

Philosophia Mathematica, 2017

Although mathematicians often use it, mathematical beauty is a philosophically challenging concep... more Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to the way we visualise them? Using a case study from graph theory (the highly symmetric Petersen graph), this paper tries to analyse aesthetic preferences in mathematical practice and to distinguish genuine aesthetic from epistemic or practical judgements. It argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract.

keywords: Cayley graphs, groups, visualizations, proofs.

Policies of inclusion of disabled students in Russia

1. Summary. One of some important changes in Russian educational system for the last decade of de... more

The impact of connectivity on the memory capacity and the retrieval dynamics of Hopfield-type networks

Most models of neural associative memory have used networks with broad connectivity. However, thi... more Most models of neural associative memory have used networks with broad connectivity. However, this seems unrealistic from a neuroanatomical perspective. A simple model of associative memory with emergent properties was introduced by Hopfield [5]. We choose this widely known model to investigate the impact of connectivity on the storage capacity and the retrieval dynamics in artificial associative networks. In this paper, we use sparse topologies where only a small fraction of possible connections were actually active. For these diluted topologies, we test different kinds of architecture; namely, randomly connected networks, regularly connected networks, small-world networks, and modular networks. Computer experiments reveal that, among these diluted topologies, the modular architectures, that exhibit a relatively high clustering coefficient and whose characteristic path lengths and total physical wiring lengths are short, produce the far better results in terms of memory storage and...

From Practice to Results in Logic and Mathematics

Aesthetic Preferences in Mathematics: A Case Study

A B S T R AC T Although mathematicians often use it, mathematical beauty is a philosophically cha... more A B S T R AC T Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory (the highly symmetric Petersen graph), this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract.

“Zooming Out and Zooming In”, Workshop "From Basic Cognition to Mathematical Practice" 19-21 September, 2016, University of Seville, Institute of Mathematics (IMUS), Spain

Searching for a different perspective on groups: some philosophical reflections

The paper offers a philosophical approach to visual representations in mathematics as leading to ... more

THOUGHT EXPERIMENTS IN MATHEMATICS

It is not news that we often make discoveries or find reasons for a mathematical proposition by t... more It is not news that we often make discoveries or find reasons for a mathematical proposition by thinking alone. But does any of this thinking count as conducting a thought experiment? The answer to that question is “yes,” but without refinement the question is uninteresting. Suppose you want to know whether the equation [8x + 12y = 6] has a solution in the integers. You might mentally substitute some integer values for the variables and calculate. In that case you would be mentally trying something out, experimenting with particular integer values, in order to test the hypothesis that the equation has no solution in the integers. Not getting a solution the first time, you might repeat the thought experiment with different integer inputs.

Volume 29, Issue 1, April 2010 Mathematical knowledge: Intuition, visualization, and understanding/Guest Edited by Leon Horsten and Irina Starikova

From Practice to New Concepts: Geometric Properties of Groups, Philosophia Scientiæ, 16 (1), 2012, pp.129-151.

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant ... more Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes.

Why do Mathematicians Need Different Ways of Presenting Mathematical Objects? The Case of Cayley Graphs

Topoi-an International Review of Philosophy, 2010

This paper investigates the role of pictures in mathematics in the particular case of Cayley grap... more This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures.

Thought Experiments in Mathematics: From Fiction to Facts

Springer eBooks, Dec 31, 2022

Like science and philosophy, thought experiments in mathematics link a problem to new epistemic r... more Like science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that were unavailable earlier. Performing an imaginary scenario can involve counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with proofs, diagrams and calculative procedures. Reflection about thought experiments helps to realize limits and opportunities of mathematical thinking. The analysis of examples suggest a broader understanding of &quot;mathematical&quot;, &quot;thought&quot; and &quot;experiment.&quot;

Thought experiments in mathematics: from fiction to facts

Like science and philosophy, thought experiments in mathematics link a problem to new epistemic r... more Like science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that were unavailable earlier. Performing an imaginary scenario can involve counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with proofs, diagrams and calculative procedures. Reflection about thought experiments helps to realize limits and opportunities of mathematical thinking. The analysis of examples suggest a broader understanding of "mathematical", "thought" and "experiment."

Policies for including disabled students in Russia

Growing attention to the issue of higher education for disabled people in Russia reflects the urg... more Growing attention to the issue of higher education for disabled people in Russia reflects the urgent and problematic character of the situation. This paper, based on current research in the field, proposes policy reforms on a regional level elaborated in the framework of the Higher Education Working Group of the International Policy Fellowship programme. Proposals include: 1. creating a service or appointing an employee responsible for the assistance of disabled students; 2. making the necessary architectural changes to ensure the accessibility of university buildings; 3. continuity of education for disabled children; 4. active efforts by universities towards communicating with potential students; 5. creating informational basis including information on rights and other relative issues for disabled persons in the sphere of higher education; 6. campaign for community awareness; 7. teachers' training on working with disabled students.

Revisiting the mutilated chessboard or the many roles of a picture

by Jean Paul Van Bendegem and Irina Starikova

The impact of connectivity on the memory capacity and the retrieval dynamics of Hopfield-type networks

Most models of neural associative memory have used networks with broad connectivity. However, thi... more Most models of neural associative memory have used networks with broad connectivity. However, this seems unrealistic from a neuroanatomical perspective. A simple model of associative memory with emergent properties was introduced by Hopfield (1982). We choose this widely known model to investigate the impact of connectivity on the storage capacity and the retrieval dynamics in Hopfield-type networks. We used sparse topologies where only a small fraction of possible connections were actually active. For these diluted topologies, we tested different kinds of architecture; namely, randomly connected networks, regularly connected networks, small-world networks, and modular networks. Computer experiments reveal that, among the diluted topologies, the modular architectures that exhibit a relatively high clustering coefficient and whose average characteristic path lengths and total physical wiring lengths are short, produce much better results in terms of pattern storage and generalization...

Thought Experiments in Mathematics

The Routledge Companion to Thought Experiments, 2017

It is not news that we often make discoveries or find reasons for a mathematical proposition by t... more It is not news that we often make discoveries or find reasons for a mathematical proposition by thinking alone. But does any of this thinking count as conducting a thought experiment? The answer to that question is "yes", but without refinement the question is uninteresting. Suppose you want to know whether the equation [ 8x + 12y = 6 ] has a solution in the integers. You might mentally substitute some integer values for the variables and calculate. In that case you would be mentally trying something out, experimenting with particular integer values, in order to test the hypothesis that the equation has no solution in the integers. Not getting a solution first time, you might repeat the thought experiment with different integer inputs. The fact that there are such mundane thought experiments is no surprise and does not answer the question we are really interested in. 2 The numerical thought experiment just given involves nothing more than applying mathematically prescribed rules (such as rules of substitution and calculation) to selected inputs. It would be more interesting if there were mathematical thought experiments in which the experimental thinking goes beyond application of mathematically prescribed rules, by using sensory imagination as a way of eliciting the benefits of past perceptual experience. 3 In what follows we will try to show that there are such thought experiments and to assess their epistemic worth. Our method will be to present some candidate thought experiments with what we hope is enough background explanation and in sufficient detail for you, the reader, to perform the relevant mental operations yourself; without this participation the paper will be neither convincing nor engaging. We have tried to avoid run of the mill 1 I would like to thank the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) and the Russian Foundation of Basic Research (RFBR). 2 For this reason we find that the category of thought experiments as characterised by Jean-Paul Van Bendegem in "Thought experiments in mathematics: anything but proof" Philosophica 72, (2003), pp. 9-33 to be too broad. 3 For a different focus, see Eduard Glas, "On the role of thought experiments in mathematical discovery" in J. Meheus and T. Nickles (eds.), Models of Discovery and Creativity, (Springer 2009). Glas says that "imagery, mental or experiential, is not essential" to the aspect of thinking that he counts as thought experiment (even when accompanied by imagery). For this reason, the kinds of thinking that we discuss in this paper do not fall under what Glas counts as thought experiment.

Picture-proofs and platonism

Croatian Journal of Philosophy, 2007

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kant... more This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the "seeing" of mathematical ideas by means of pictures, diagrams, thought experiments, etc. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss -James Robert Brown's recent book Philosophy of Mathematics, in particular, his primarily examples and analogies. I then consider some alternatives and counterarguments, namely John Norton's opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification,...

From Practice to New Concepts: Geometric Properties of Groups

Philosophia Scientiae, 2012

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant ... more Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation

Aesthetic Preferences in Mathematics: A Case Study†

Philosophia Mathematica, 2017

Although mathematicians often use it, mathematical beauty is a philosophically challenging concep... more Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to the way we visualise them? Using a case study from graph theory (the highly symmetric Petersen graph), this paper tries to analyse aesthetic preferences in mathematical practice and to distinguish genuine aesthetic from epistemic or practical judgements. It argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract.

keywords: Cayley graphs, groups, visualizations, proofs.

Policies of inclusion of disabled students in Russia

1. Summary. One of some important changes in Russian educational system for the last decade of de... more

The impact of connectivity on the memory capacity and the retrieval dynamics of Hopfield-type networks

Most models of neural associative memory have used networks with broad connectivity. However, thi... more Most models of neural associative memory have used networks with broad connectivity. However, this seems unrealistic from a neuroanatomical perspective. A simple model of associative memory with emergent properties was introduced by Hopfield [5]. We choose this widely known model to investigate the impact of connectivity on the storage capacity and the retrieval dynamics in artificial associative networks. In this paper, we use sparse topologies where only a small fraction of possible connections were actually active. For these diluted topologies, we test different kinds of architecture; namely, randomly connected networks, regularly connected networks, small-world networks, and modular networks. Computer experiments reveal that, among these diluted topologies, the modular architectures, that exhibit a relatively high clustering coefficient and whose characteristic path lengths and total physical wiring lengths are short, produce the far better results in terms of memory storage and...

From Practice to Results in Logic and Mathematics

Aesthetic Preferences in Mathematics: A Case Study

A B S T R AC T Although mathematicians often use it, mathematical beauty is a philosophically cha... more A B S T R AC T Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory (the highly symmetric Petersen graph), this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract.

“Zooming Out and Zooming In”, Workshop "From Basic Cognition to Mathematical Practice" 19-21 September, 2016, University of Seville, Institute of Mathematics (IMUS), Spain

Searching for a different perspective on groups: some philosophical reflections

The paper offers a philosophical approach to visual representations in mathematics as leading to ... more

THOUGHT EXPERIMENTS IN MATHEMATICS

It is not news that we often make discoveries or find reasons for a mathematical proposition by t... more It is not news that we often make discoveries or find reasons for a mathematical proposition by thinking alone. But does any of this thinking count as conducting a thought experiment? The answer to that question is “yes,” but without refinement the question is uninteresting. Suppose you want to know whether the equation [8x + 12y = 6] has a solution in the integers. You might mentally substitute some integer values for the variables and calculate. In that case you would be mentally trying something out, experimenting with particular integer values, in order to test the hypothesis that the equation has no solution in the integers. Not getting a solution the first time, you might repeat the thought experiment with different integer inputs.

Volume 29, Issue 1, April 2010 Mathematical knowledge: Intuition, visualization, and understanding/Guest Edited by Leon Horsten and Irina Starikova

From Practice to New Concepts: Geometric Properties of Groups, Philosophia Scientiæ, 16 (1), 2012, pp.129-151.

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant ... more Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes.

Why do Mathematicians Need Different Ways of Presenting Mathematical Objects? The Case of Cayley Graphs

Topoi-an International Review of Philosophy, 2010

This paper investigates the role of pictures in mathematics in the particular case of Cayley grap... more This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures.

Visual Representation and Reasoning

Quotations about Mathematics

Reasoning with Cognitive Representations