The aim of this paper is to present two arguments for an approach that does not radically differentiate the methodology of mathematics from the methodology of other sciences (especially empirical ones), and that may be more effective or natural for mathematical practice.

A the beginning, the approach of Halina Mortimer [1], [2], one of the representatives of the Lviv-Warsaw School, who was concerned with the logic of induction, i.e. the methodology for developing the theory of empirical sciences, is presented,. Although Mortimer appreciated some of the solutions of Carnap and Hintikka in this area [3], [4], [5], she was of the opinion that they were too idealised. As an alternative, she proposed the solution of Henry Kyburg [6], who pointed out the connection between the system of inductive logic and a given empirical theory through mathematical probability. This proposal did not allow logic and mathematics to be treated as a priori theories explaining the reality described by the empirical sciences, but rather as tools for specifying and ordering these theories.

We will try to apply the above position to the analysis of two questions. First, we will comparatively analyse the methodology of Dedekind [7], [8], [9], [10], who constructed some mathematical models, and the methodology of Cantor [11], [12], [13], who used the genetic-deductive method. This may have been due to Cantor’s idealised Platonic assumptions about mathematics. This sometimes led to errors in his work, which were corrected by later communities of mathematicians. Dedekind, approached mathematics in a more empirical way. He also produced more effective solutions.

We also note the scientific environments that might have influenced the mathematicians. Dedekind’s early environment included mathematicians such as Gauss and Dirichlet, while Cantor’s included Weierstrass and Kronecker. Gauss was also known as an experimental physicist, while Weierstrass devoted himself exclusively to detailed work on mathematical problems [14].

Secondly, we will analyse for comparative purposes the mathematical model that Dedekind built in the Foundations of Mathematics (N) and the modern flow model for a social network [15], [16]. We will pay attention to common and different aspects that could (or might) accompany the creation or recreation of these models.

For this purpose, we will present a proposal for a general cognitive scheme for building a mathematical model in individual mathematical practice. Based on the definition of the model [17], we will use the proposal of Giaquinto [18], [19], who, following Kosslyn, shows how a structure can be “grasped” visually.

The conclusions of these considerations point to the need to pay more attention to the individual and social context of scientific discovery. This would allow us to show more generally that mathematical knowledge, as a product of individual and social practice, is not necessarily developed in a continuous way, where continuity is ensured by the use of the genetic-deductive method. New theoretical structures can also be constructed in relation to - broadly understood - experience, and not only in strict relation to current theory.

In the last decades, there has been a fiery debate about computer-assisted proofs (CAPs). Started by Appel & Haken’s (1977) proof of the four-color theorem and the mixed reception it received, the debate centers around two issues. Let t be a mathematical theorem such that there is a CAP for t, but no ordinary (i.e. non-computational) proof. Firstly, is t merely verified by the CAP or does the CAP amount to a full proof (Rota 1997)? Secondly, is t known a posteriori because there is no proof for t that could be surveyed (Tymozko 1979, Kitcher 1984, McEvoy 2007)?

In the talk, I’ll contribute to the debate in two ways. Firstly, I’ll report initial findings from an ongoing data-driven approach that aims to analyze articles published on the ArXiv to understand how, and to what extent, mathematicians rely on CAPs. Results may indicate whether the mathematical community takes CAPs to be on par with ordinary proofs or not.

Secondly, I’ll argue that the justification provided by a CAP is different in degree, but not in kind, from the justification provided by an ordinary proof. If the latter is taken to yield a priori justification, the former does provide a priori justification, too. The argument relies on the idea that a CAP can be translated into an ordinary proof. While the translated proof is notoriously complex and probably impossible to survey by a single human agent, these differences are insufficient to claim that the CAP provides merely an a posteriori justification.

References

• Appel, Kenneth, and Wolfgang Haken. 1977. “Every Planar Map is Four Colorable. Part I: Discharging.” Illinois Journal of Mathematics 21 (3): 429–90.
• Appel, Kenneth, Wolfgang Haken, and John Koch. 1977. “Every Planar Map is Four Colorable. Part II: Reducibility.” Illinois journal of Mathematics 21 (3): 491–567.
• Kitcher, Philip. 1984. The Nature of Mathematical Knowledge. New York: Oxford University Press. https://doi.org/10.1093/0195035410.001.0001
• McEvoy, Mark. 2007. “The Epistemological Status of Computer-Assisted Proofs.” Philosophia Mathematica 16 (3): 374–87. https://doi.org/10.1093/philmat/nkn014.
• Rota, Gian-Carlo. 1997. “The Phenomenology of Mathematical Proof.” Synthese 111 (2): 183–96. https://doi.org/10.1023/A:1004974521326.
• Tymoczko, Thomas. 1979. “The Four-Color Problem and Its Philosophical Significance.” The Journal of Philosophy 76 (2): 57–83. https://doi.org/10.2307/2025976.

In the Call for Abstracts (CfA) for the Workshop of the GDWP questions are raised concerning the notion of scientific practices as there is still no common definition. Mostly, the notion of scientific practices is still linked to laboratories, in which the works of Bruno Latour or Karin Knorr-Cetina took place and thrived in the 1970s and 1980s. (Concerning the special institutional frame of the works of Bruno Latour see, e.g., Lars GERTENBACH / Henning LAUX, Zur Aktualität von Bruno Latour. Einführung in sein Werk (Wiesbaden 2019), 87-88 and Matthias WIESER, Das Netzwerk von Bruno Latour. Die Akteur-Netzwerk-Theorie zwischen Science & Technology Studies und poststrukturalistischer Soziologie (Bielefeld 2012), 5. And concerning the notion of laboratories as primary places of early practice studies see for example WIESER, Das Netzwerk von Bruno Latour, 22 or 24.) But as such institutions are only one of many, where scientific practices emerge and are at work, it becomes evident that practices in other places are seldom acknowledged. One of them is universities – scientific institutions combined with an emphasis on higher education at the same time. If we think of universities as scientific institutions, what are scientific practices that we come across there? I would argue that teaching would be one answer, next to doing research, writing books and essays, etc., which already shows that scientific practices are different from those applied in laboratories.

However, it seems, that there is still no sufficient theoretical background for research on scientific practices in universities, such as, e.g., teaching, although this practice is often (but not always) shaped by the newest state of research and researchers themselves. At universities (ideally) knowledge is discussed and exchanged (mostly with a hierarchical stance between lecturer and students). So, if discussions on the newest research do take place in classes or lectures, science can be debated and enhanced in those very moments, which is, as I would argue, part of scientific practices as well.

Also, adding the historical perspective as another argumentative layer, in the case of, e.g., Austria it becomes even more evident that universities and most of all, university teaching, were both a big part of “science”. For example, especially since the Student Revolution of 1848/49, “science” was embedded in Austrian lecture halls and textbooks and embedded into universities. (See for example Thomas MAISEL, Lehr- und Lernfreiheit und die ersten Schritte zu einer Universitäts- und Studienreform im Revolutionsjahr 1848, in: Christof AICHNER, Brigitte MAZOHL (Hg.), Die Thun-Hohenstein’schen Universitätsreformen 1849-1860 (Wien / Köln / Weimar 2017), 99-117, 105-106.) Consequently, in terms of pre-20th-Century science, it becomes evident that the term “scientific practices” only linked to non-university institutions, such as laboratories, becomes narrow, and thus, scientific practices at universities (e.g., university teaching) and their specific mechanisms (1) should be included when we think of “scientific practices” and (2) should be investigated further in the context of the “practice turn”.

When we talk about scientific practice do we describe behaviour, or do we prescribe what behaviour ought to be? Neither is sufficient. Scientific practice cannot be ideal behaviour, because then we have never encountered it. Nor can it be understood as a historical list of all scientific actions. It is something we will do tomorrow, too.

In this paper, I argue that when we talk about scientific practice we talk about:

• actual systems of practices,
• that have been retained because they have certain features,
• which help us learn about the world.

I articulate this view and defend it against five related criticisms.

This a novel application of a selectionist teleological account (inspired by C.S. Peirce but developed over decades by T.L. Short). To give a sketch, scientific practices are actual, particular things with many features, but they have been retained because some of those features allow us to achieve a certain type of outcome. This type is the “final cause” of the practice.

Final causes are not strange efficient causes (e.g., backwards causation or the result of an omnipresent will). They are types of outcome that explain why certain actual traits were selected. “To tell time” answers the questions “why are these parts of a clock are assembled like this?” and “why did people build such things?”. Not every characteristic of scientific practice is explained by science’s aim, no more than every characteristic of a clock is explained by telling time.

Final causes do not require intentionality. No creature intentionally decided to retain retinal for the purpose of increasing visual acuity, yet that is why it was retained. A societal structure that provided free time enough for country priests to develop new experimental methods was not intended to develop such methods. Final causation explains why we have retained remnants of such methods, while new labs need not include parishes.

Rowbottom (2023) gives five arguments critical of this view, namely: Scientists (1) do not have a collective purpose; (2) are not directed by above to any end; (3) cannot be guided by an ideal (and are in fact often better off when acting non-ideally); (4) do not share any characteristic that could be called their aim; (5) and when practices are introduced for a purpose, they often do not serve that purpose, which is instead satisfied in multiple ways.

I address each with the pragmatist approach. As mentioned: intentionality (collective or otherwise) is not required, nor is a guiding will. A pluralism of practices is encouraged because increased variety provides more to select from.

The practice turn has provided much excellent work regarding how philosophers should understand practice. This paper aims to assist such work by providing a criterion for when such practices are scientific. But what is the aim of science? Answering this is also the task of science (including philosophy). Vaguely, it is to learn about the world. Philosophers provide more specific answers. Many have been and should be put forward. Only then can we learn about why some are retained.

In this presentation I aim to show that Newton’s experimental philosophical methodology exhibits a salient modal-explorative nature (cf. Steinle 2006/2016, Gelfert 2016, Massimi 2019). First, I shall argue that Newton’s methodology of queries can be better qualified as being explorative in a modal sense (cf. Newton 1959-1977, 1713/1999, 1717/1952), i.e., Newton provides under the label of queries various scientific models with non-actual/fictional target-system (e.g., aether models with low resistance from his “Queries” to the Opticks). Queries have the methodological role of suggesting further experiments and advancing theoretical points for extending the boundaries of Newton’s existing knowledge. Via his aether models, Newton aims to find out plausible possible causal mechanisms for the production of gravitation.

Second, this modal-explorative methodology has immediate consequences over his metaphysics of laws of nature. Recent commentators (e.g., Biener & Schliesser 2017, Walsh 2017, Henry 2020, Hazelwood 2023) remarked that Newton’s definition of active principles as laws, causes, natures, or dispositions gives rise to various conceptual tensions. However, I aim to show that the tension inherent in his conception of laws is merely apparent since he is wrestling with multiple possibilities of making a theory of laws to work. According to the advanced modal-explorative reading, Newton attempts to find out a plausible contender for what laws of nature may be. Relative to the latter reading, his definition of active principles in terms of laws and dispositions should not be taken at face value, but seen as a “never-at-rest” explorative activity.