David Bloor et la contingence des sciences formelles // David Bloor and the contingency of the formal sciences

 

Andrew ARANA, Archives Henri-Poincaré—PReST (UMR 7117), Université de Lorraine

Contingency in The Boundaries of Mathematical Disciplines

How are the boundaries of mathematical disciplines drawn? How do we distinguish arithmetic from geometry, or topology from analysis? Are the boundaries between these disciplines part of how mathematical nature is carved at the joints, as the inevitabilist would have it? Or is their drawing an activity of temporally-situated agents responding to their material and scientific conditions, as the contingentist would have it? In this talk, I will discuss these questions.

 

Michael J. BARANY, University of Edinburgh

The Contingency of Remedial Mathematics

A critical rhetorical and conceptual feature of Bloor's contingentist account of mathematical knowledge was a social theory of taken-for-granted examples of elementary mathematics. It is well known that elementary mathematics has a complex and consequential relationship to mathematics and formal sciences writ large, often confounding or misleading. I will develop a small but significant twist to Bloor's approach that reframes the relationship between elementary and less-elementary mathematics by focusing on the situated social production of epistemic closure. In particular, social processes of remediation can be identified as constitutive points of practical and conceptual continuity across different forms and contexts of mathematics. Bloor identified the social contexts of agreeing to the unicity and even self-evidence of mathematical claims as critical sites for the manifestation of social power and interests. A remedial view extends this insight by understanding closure to be a perpetually provisional and unfinished facet of mathematical epistemology. This permits a contingentist account of the relationship between settled and creative mathematics, while also accounting for the social production and acceptance of appearances of mathematics to be inevitable. I relate these analyses to Bloor's empiricist commitments and his understanding of formality, informality, and the relationship between them.

 

Jean Paul van BENDEGEM, Vrije Universiteit Brussel – Centre Leo Apostel (CLEA)

The Singular and the Plural in the Philosophy of Mathematics
Many philosophers and mathematicians will agree that changing the focus from mathematical theories, especially foundational theories, to what mathematicians do in their daily work has been an important development. There is, however, a highly curious feature that may seem quite innocent but, as I will argue, is extremely important, and that is that the label for this development uses the singular: the history and philosophy of mathematical practice. Not practices (as I prefer and claim it should be). My defence of the plural consists of another look at (a) one of the favourite examples of David Bloor, namely modus ponens, (b) the role of contradictions, calling in Ludwig Wittgenstein (and thereby Bloor as well), and (c) the status of infinity, a concept that in itself needs the plural. Present in the background, Graham Priest’s dialetheism – the view that there are true contradictions in the world itself, not merely in our theories – will serve as a test case, as it represents a rather extreme position (though Priest himself probably would stick to the singular).

 

Paul ERNEST, University of Exeter

David Bloor's Philosophy of Mathematics and the Beliefs of Working Mathematicians
A key tenet of David Bloor's philosophy of mathematics is that the beliefs of mathematicians must be explained, even if they correspond to the widely accepted dogmas of the absolute certainty of mathematical knowledge and of the objective existence of mathematical objects (Platonism). Bloor argues that explanations of such beliefs should be (1) causal, concerned with the conditions which bring about the beliefs; (2) impartial with respect to their truth and falsity; (3) symmetric in explaining both true and false beliefs.
This paper argues that these epistemic and ontic dogmas, irrespective of their truth or falsity, can be partly explained by their widespread acceptance as values. The corresponding beliefs should be seen as located in axiology and not epistemology or ontology. Mathematicians are recruited into the social practices of mathematics, and these come with established modes of working as well as metamathematical beliefs and values.
However, the claim is not that these dogmas are near-arbitrary; held in place solely by institutional decisions. There is a strong rational basis for these beliefs and values, because they are either true or nearly true. Mathematical knowledge is well justified because of proofs and successful applications. Mathematical objects are enduring entities that preexist the formation of mathematicians and continue in their existence beyond them. However, the objects of mathematics can be conceived as cultural instead of Platonic entities. Likewise, mathematical knowledge endures but need not be understood as based on metaphysical necessity. It can also be seen as based on social institutions imposing necessity, based on a deontological ‘must’.
Bloor’s arguments about the explanations of ontic and epistemic beliefs are independent of the epistemological and ontological assumptions of the status of mathematical knowledge and the nature of mathematical objects.    

 

Bart van KERKHOVE and Sander POULIART, Centre Leo Apostel, Vrije Universiteit Brussel

Bloor vs Heintz vs Restivo on and in the Sociology of Mathematics

This paper will delve into the contributions of two lesser-known proponents of a social theory of mathematical knowledge. We shall put their accounts in the wider context of the field, compare their basic tenets to those of David Bloor so as to position them towards it, and then also assess whether and to what extent they give or leave more or less room for contingency in mathematics.

The first of these alternative accounts is that by the German scholar Bettina Heintz. Flowing from systems theory, it aspires to bridge contrasting sociological and philosophical concerns by formulating a general socio-historical explanation for the apparent universality and inevitability with which mathematical knowledge comes to us, through proof. At its core stands the idea that proof production is not mainly aimed at establishing certainty, but rather at facilitating proper communication, namely with a view to enhancing the badly needed trust in mathematical results. This position is better understood once its pragmatic nature is accepted, as what above all seems to count for working mathematicians is whether or not purported results can be relied upon for further research.

The second sociological account that we shall look into is that by the American sociologist Sal Restivo. Borrowing a core idea of the controversial German social critic Oswald Spengler, namely that different cultures are utterly incommensurable, and, accordingly, also have their very own conception of number, Restivo has developed a theory of 'mathematical sociologism'. It essentially says that mathematical production requires the use, and prior or simultaneous development, of conceptual tools, whereby one adopts learnt techniques as well as personal skills essentially as a member of different social groups. As a result, all mathematical 'things', like non-material things or things experienced in general (including talk, persons, minds, cognition) are constitutively social.

 

Franci MANGRAVITI, ETH Zurich

In Defense of the Notion of Alternative Mathematics

The philosophy of mathematical practice has done much work to unearth ways in which much of the standard picture of mathematics as a uniform monolith is built on questionable assumptions, simplifications, and erasures. Furthermore, the expressiveness of set theory and the rise of metamathematical methods have made it seemingly possible to encompass all kinds of alternatives within a single framework. Together, these two factors put pressure on the very notion of alternativeness - and thus, on the idea that mathematics could be contingent - not, as in the old days, from the perspective of there only being one ‘right’ mathematics, but rather from the perspective of mathematics already being as ‘open’ as possible, and a focus on alternatives unwarrantedly contributing to old-fashioned stereotypes. In this paper, I push back against this by arguing that the very possibility of constructive structural critiques of mathematics depends on alternatives being conceived in a stronger sense than the liberal and reductionist narratives can allow.

 

Baptiste MÉLÈS, CNRS, Archives Henri-Poincaré—PReST (UMR 7117), Université de Lorraine

Do Computer-Assisted Proofs Change the Game for Mathematical Contingentism?

Far from being a secondary point in David Bloor's argument, contingentism in mathematics is its cornerstone. One might think that the mechanization of mathematical proofs by proof assistants has changed the game by giving mathematics a new form of necessity. However, we will see that Bloor's arguments have both as much and as little reason to hold: other forms of mathematics are still possible, but sociological empiricism is still not proven.

 

José Antonio PÉREZ-ESCOBAR, Department of Logic, History and Philosophy of Science, UNED, Madrid
Colin Jakob RITTBERG, Centre Leo Apostel, Vrije Universiteit Brussel

Petrification in Contemporary Set Theory: The Multiverse and the Later Wittgenstein
As Bloor’s contingentist sociology of mathematics — rooted in Wittgenstein’s later philosophy — reminds us, choices in mathematics are historically situated. In this talk we follow up on the idea that these choices, which are contingent by nature, can solidify into normative demands. We argue that Wittgenstein’s notion of petrification can be used to explain this solidification. We focus on phenomena in advanced mathematics, thereby extending the debate about mathematical petrification by pushing beyond its current diet of examples of a very basic mathematical nature. Second, we analyze current disagreements on the absolute undecidability of CH under the notion of petrification. We make use of hinge epistemology to argue that contemporary construction techniques for set-theoretic models in which the Continuum Hypothesis holds and those in which it fails have petrified into the normative demand that CH remain undecidable. This argument draws on and enriches the arguments put forth by set theorist Joel David Hamkins.

 

Léna SOLER, Archives Henri-Poincaré—PReST (UMR 7117), Université de   Lorraine

Bloor, Pioneer of Contingentism. Inevitabilist Strategies about Mathematics and Logic, and their Analogues in the Natural Sciences
This reflection pursues a threefold objective. First, to reconstruct David Bloor’s position on mathematics and logic as a form of “contingentism” (C), opposed to “inevitabilism” (I), in the senses introduced by Hacking in philosophy of science at the turn of the 2000s. Bloor can naturally be reconceptualized as a radical contingentist about science, including about what is usually regarded as the “realm of necessity”, namely mathematics and logic, especially proofs and results. Under this reconceptualization, we realize that already in the first 1976 edition of Knowledge and Social Imagery, Bloor had identified and masterfully discussed most of the conceptual and argumentative difficulties of the I/C issue, including many inevitabilist strategies commonly employed to dismiss or neutralize candidates for alternative mathematics – thereby maintaining the standard status of mathematics as inevitable, if not necessary. This invaluable Bloorian contribution, however, has gone largely unheeded by philosophers of science, including philosophers of mathematics. While the “strong programme” has been widely debated in relation to the natural sciences, the Bloorian contingentist analyses of mathematics and logic contained in it have, for their part, largely been treated as ‘nonexistent’.
Against this background, I will begin by specifying the main features of Bloor’s contingentism. Then, I will characterize in greater detail the main inevitabilist strategies directed at ingredients of mathematics and logic that Bloor systematically identified and critically discussed – this is my second objective. My third will be to compare the previous strategies, as applied to the formal sciences, with the inevitabilist strategies applied to the empirical sciences that have been brought to light in the post-Hacking literature devoted to the I/C debate—a debate that has so far been almost exclusively focused on the natural sciences while neglecting the formal ones. The comparison will reveal striking structural analogies, thereby showing the transdisciplinary character of the strategies Bloor had singled out at an early stage.

 

Roy WAGNER, ETH Zurich

Contingency in the History of Euclidean Geometry
In current historiography, attempted historical proofs of the parallel postulate are presented as inevitably wrong due to the existence of non-Euclidean models obeying Euclid's other axioms. I argue that this approach is committed to an inevitabilist view of mathematics and that under a suitable historical interpretation, some of these proofs may be as good as many well-established Euclidean proofs. I then argue that this historical interpretation articulates a contingent alternative mathematics that is irreducibly different and as good as current mathematics. I conclude by framing both current and Greek geometries as contingent on their distinct socio-cultural and institutional historical circumstances. The upshot is that mathematics is no less contingent than the underlying historical circumstances.